We know from New Who that the TARDIS was a "museum piece" when the Doctor stole (erm, I mean "borrowed") her, which suggests that she may predate the Doctor himself. But are there any canonical statements of her actual age, and are there any canonical records of her history from before she met the Doctor?
Age: Several thousand years. Little history known.
The Doctor's TARDIS is identified many times as a Type Forty. Type Forties are indeed referred to several times as being obsolete. As early as the 3rd Doctor story The Claws of Axos (Classic Season 8, Serial 3), the Master refers to it as a museum piece.
(The Master has tidied up all the wiring and is trying to get the old girl started.)
MASTER: Oh, hopeless! Overweight, under-powered old museum piece!
(A feeble dematerialisation sound starts up but the time rotor stays still.)
MASTER: Ah, no proper stabiliser. Oh well, let's try again.
(This time there is a small explosion.)
MASTER: You may as well try to fly a second hand gas stove!
The Claws of Axos. Transcript from Chakoteya.net
When the 4th Doctor visits Gallifrey in The Deadly Assassin (Classic Season 14, Serial 4), we learn that 305 Type 40s had been registered. All but one had been deregistered, the last one was removed by a Malfeasance Tribunal (the "malfeasance" presumably being the Doctor's).
SPANDRELL: Data retrieval. Request information on all Type Forty TT capsules currently operational.
COMPUTER: Negative information. Type Forty TT capsules are deregistered and non-operational.
SPANDRELL: Report number of de-registrations.
COMPUTER: Three hundred and four.
SPANDRELL: Report number of registrations.
COMPUTER: Three hundred and five.
SPANDRELL: Report reason for numerical imbalance.
COMPUTER: One capsule removed from register. Reference, Malfeasance Tribunal order dated three zero nine nine zero six.
The Deadly Assassin. Transcript from Chakoteya.net
In the unbroadcast 4th Doctor story Shada (which would have been Classic Season 17, Serial 6), we meet a retired Time Lord called Professor Chronotis who is apparently several thousand years old. The Professor considers the Type Forty to be about as old as himself.
The Professor peered over his spectacles at the police box. 'Yes of course. It's a Type 40 TARDIS, isn't it?'
Romana opened the door of the TARDIS. 'Yes. They were on the Vintage and Veteran Vehicles syllabus at the Academy. It's amazing this one is still going.'
'I remember when they first came out, you know. When I was just a boy.' He chuckled and reached out, stroking the wooden shell with affection. 'That'll show you how old I am. A vintage veteren myself.'
Romana leaned down and tweaked the Professor's nose. 'Nonsense. As they say nowadays on Gallifrey, '6,000 is the new 4,000.'
Shada: Script by Douglas Adams. Novelisation by Gareth Roberts
This suggests that the Type Forty appeared at least 6,000 years earlier (in the Professor's timeline). At least one had not been deactivated by the time the Doctor stole it. In fact, the scene in The Name of the Doctor that shows the 1st Doctor stealing the TARDIS shows a row of identical TARDISs and Clara tells the doctor which one to pick.
CLARA: Sorry, but you're about to make a very big mistake. Don't steal that one, steal this one. The navigation system's knackered, but you'll have much more fun.
The Name of the Doctor. Transcript from Chakoteya.net
I'm not aware of any earlier information about the Doctor's TARDIS.