What was the exact age Darth Vader was when he finally dies at the end of Return of the Jedi?
The best info available suggests that he was 45 when he died.
To determine this, we need to use a few sources. Note that some of these are no longer considered fully canon.
According to this post by Leland Chee, Anakin was born in 41.9 BBY.
Based on various cast interviews, we know that Anakin was aged 9 in Episode I (and that this film takes place in 32 BBY).
According to the Del Rey New Essential Chronology the Battle of Endor takes place in 4.1 ABY
That means Vader/Anakin died .7 of a standard Galactic Year (approx 8 months) before his 46th birthday. Even taking into account that a galactic standard year is longer than an Earth year (368 days versus 365), there's not enough of a different to be important to our calculations.
Lets Do The Math
So we know that he was Age 9 during The Phantom Menace.
And we know from the conversation with Anakin and Obi-Wan in the turbolift at the beginning of Attack of the Clones that it takes place 10 years later
Obi-Wan: Calm down Anakin.
Anakin: Sorry master, I'm just nervous I haven't seen her in ten years.
That puts him at 19 when the Clone Wars start.
Another 3 years pass in between Episode II and III putting him at 22.
After that we know that Luke was 19 in A New Hope putting Vader at 41.
Another 3 years pass Putting Luke at 22 and Vader at 44.
After that only one more year passes until Epidode Six putting Luke at 23 and Vader at 45 years when he died.