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We have a few clues that would suggest that Gurgeh is in his mid-60's.

Special Circumstances seem to have been keeping the Azadian Empire secret since they (allegedly) learned of the Azadian Empire some 74(ish) years ago.

When Gurgeh realises that he was played as a patsy by Special Circumstance, he then heads off on a paranoid tangent, asking whether he was bred by SC for the specific purpose of becoming the top Azad player of all time, thus enabling SC to defeat the Empire from the inside-out.

'And when did you first get that sniff of contact?' Gurgeh said, turning to the drone. 'How long have you been sitting on this?'
The drone hesitated. 'Not long,' it said eventually. 'Seventy-three years.'
'You people certainly don't rush into things, do you?'
'Only when we've no choice,' the drone agreed.

and

'All my life,' Gurgeh said quietly, looking past the drone to the dull, dead landscape outside the tall windows. 'Sixty years… and how long has the Culture known about the Empire?'
'About - ah! You're thinking we shaped you somehow. Not so. If we did that sort of thing we wouldn't need outsider "mercenaries" like Shohobohaum Za to do the really dirty work.'

We also have this from Gurgeh, confirming that he's no older than 60-70.

'I have learned to recognise the signs,' the professor said. She was twice Gurgeh's age, well into her second century, but still tall and handsome and striking. Her skin was pale and her hair was white, as it always had been, and cropped. 'Another of my students humiliated?'

We have a few clues that would suggest that Gurgeh is in his mid-60's.

Special Circumstances seem to have been keeping the Azadian Empire secret since they (allegedly) learned of its existence some 74(ish) years ago.

When Gurgeh finally realises that he was played as a patsy by Special Circumstance, he then heads off on a paranoid tangent, asking whether he was bred by SC for the specific purpose of becoming the top Azad player of all time, thus enabling SC to defeat the Empire from the inside-out. This would strongly imply that his age is congruent with the point at which SC decided that 'something needs to be done'.

'And when did you first get that sniff of contact?' Gurgeh said, turning to the drone. 'How long have you been sitting on this?'
The drone hesitated. 'Not long,' it said eventually. 'Seventy-three years.'
'You people certainly don't rush into things, do you?'
'Only when we've no choice,' the drone agreed.

and

'All my life,' Gurgeh said quietly, looking past the drone to the dull, dead landscape outside the tall windows. 'Sixty years… and how long has the Culture known about the Empire?'
'About - ah! You're thinking we shaped you somehow. Not so. If we did that sort of thing we wouldn't need outsider "mercenaries" like Shohobohaum Za to do the really dirty work.'

We also have this from Gurgeh, confirming that can't be older than 60-70 if his colleague is less than 150.

'I have learned to recognise the signs,' the professor said. She was twice Gurgeh's age, well into her second century, but still tall and handsome and striking. Her skin was pale and her hair was white, as it always had been, and cropped. 'Another of my students humiliated?'

Gurgeh seems to be in his mid-60's. Special Circumstances seem to have been keeping the Azadian Empire secret since they (allegedly) learned of the Azadian Empire some 74(ish) years ago.

When Gurgeh realises that he was played as a patsy by Special Circumstance, he then heads off on a paranoid tangent, asking whether he was bred by SC for the specific purpose of becoming the top Azad player of all time, thus enabling SC to defeat the Empire from the inside-out.

'And when did you first get that sniff of contact?' Gurgeh said, turning to the drone. 'How long have you been sitting on this?'
The drone hesitated. 'Not long,' it said eventually. 'Seventy-three years.'
'You people certainly don't rush into things, do you?'
'Only when we've no choice,' the drone agreed.

and

'All my life,' Gurgeh said quietly, looking past the drone to the dull, dead landscape outside the tall windows. 'Sixty years… and how long has the Culture known about the Empire?'
'About - ah! You're thinking we shaped you somehow. Not so. If we did that sort of thing we wouldn't need outsider "mercenaries" like Shohobohaum Za to do the really dirty work.'

and

'I have learned to recognise the signs,' the professor said. She was twice Gurgeh's age, well into her second century, but still tall and handsome and striking. Her skin was pale and her hair was white, as it always had been, and cropped. 'Another of my students humiliated?'

We have a few clues that would suggest that Gurgeh is in his mid-60's. 

Special Circumstances seem to have been keeping the Azadian Empire secret since they (allegedly) learned of the Azadian Empire some 74(ish) years ago.

When Gurgeh realises that he was played as a patsy by Special Circumstance, he then heads off on a paranoid tangent, asking whether he was bred by SC for the specific purpose of becoming the top Azad player of all time, thus enabling SC to defeat the Empire from the inside-out.

'And when did you first get that sniff of contact?' Gurgeh said, turning to the drone. 'How long have you been sitting on this?'
The drone hesitated. 'Not long,' it said eventually. 'Seventy-three years.'
'You people certainly don't rush into things, do you?'
'Only when we've no choice,' the drone agreed.

and

'All my life,' Gurgeh said quietly, looking past the drone to the dull, dead landscape outside the tall windows. 'Sixty years… and how long has the Culture known about the Empire?'
'About - ah! You're thinking we shaped you somehow. Not so. If we did that sort of thing we wouldn't need outsider "mercenaries" like Shohobohaum Za to do the really dirty work.'

We also have this from Gurgeh, confirming that he's no older than 60-70.

'I have learned to recognise the signs,' the professor said. She was twice Gurgeh's age, well into her second century, but still tall and handsome and striking. Her skin was pale and her hair was white, as it always had been, and cropped. 'Another of my students humiliated?'

Gurgeh seems to be in his mid-60's. This is confirmed when he realises that Contact have been keeping the Azadian Empire secret since it's initial discovery 74(ish) years ago. When he finally learns that he was played as a patsy for Special Circumstance, he then asks whether he was bred by SC for the specific purpose of becoming the top game player of all time, playing Azad and defeating the Empire.

'And when did you first get that sniff of contact?' Gurgeh said, turning to the drone. 'How long have you been sitting on this?'
The drone hesitated. 'Not long,' it said eventually. 'Seventy-three years.'
'You people certainly don't rush into things, do you?'
'Only when we've no choice,' the drone agreed.

and

'All my life,' Gurgeh said quietly, looking past the drone to the dull, dead landscape outside the tall windows. 'Sixty years… and how long has the Culture known about the Empire?'

Suffice to say, the drone's response is emphatic that they didn't.

Gurgeh seems to be in his mid-60's. Special Circumstances seem to have been keeping the Azadian Empire secret since they (allegedly) learned of the Azadian Empire some 74(ish) years ago. 

When Gurgeh realises that he was played as a patsy by Special Circumstance, he then heads off on a paranoid tangent, asking whether he was bred by SC for the specific purpose of becoming the top Azad player of all time, thus enabling SC to defeat the Empire from the inside-out.

'And when did you first get that sniff of contact?' Gurgeh said, turning to the drone. 'How long have you been sitting on this?'
The drone hesitated. 'Not long,' it said eventually. 'Seventy-three years.'
'You people certainly don't rush into things, do you?'
'Only when we've no choice,' the drone agreed.

and

'All my life,' Gurgeh said quietly, looking past the drone to the dull, dead landscape outside the tall windows. 'Sixty years… and how long has the Culture known about the Empire?'
'About - ah! You're thinking we shaped you somehow. Not so. If we did that sort of thing we wouldn't need outsider "mercenaries" like Shohobohaum Za to do the really dirty work.'

and

'I have learned to recognise the signs,' the professor said. She was twice Gurgeh's age, well into her second century, but still tall and handsome and striking. Her skin was pale and her hair was white, as it always had been, and cropped. 'Another of my students humiliated?'