There is no clear established canon for how old the First Doctor was when he first appeared on TV, but common speculation places his age somewhere around 400-500 years old.
The main issue of contention is that there are too many official conflicting sources. He was in his 890s, and his 440s, and his 250s, and his 120s depending on which source you believe.
Things are even more muddled by the fact that at his death he'd been keeping a 500 year diary and since he probably didn't start it as a toddler he was probably 550-600 at the time of his death. The simplest and most logical answer is that he simply doesn't know and hasn't for a long time which would explain all the differing ages he's given over the years.
In the 1968 Doctor Who Annual, the article "Phoenix in the TARDIS"—the first official thing ever written about regeneration—said that the First Doctor was around 900 years old when he regenerated.
The Second Doctor gave his age as 450 in Tomb of the Cybermen. Until that point, he'd apparently been traveling with human companions continuously since his regeneration, and none of them seemed to have aged any extra decades, so we can assume that the First Doctor was in his 440s.
According to what the Fourth Doctor told Romana in "The Ribos Operation" and "The Pirate Planet", he was only 236 when he fled Gallifrey with Susan in his stolen TARDIS. Given Susan's age, there's no way they could have traveled for 650 years together, or 200. In fact, it's implied that they only gallivanted around spacetime for a short time before coming to Earth in 1963, which would put the First Doctor somewhere in his 240s. However, we know from Ribos that by this point the Doctor was habitually shaving a few years off his age, so maybe it was more like his 250s.
Both "Time Crash" and an RTD-era Doctor Who Annual article imply that the Doctor was barely out of adolescence when he left Gallifrey. RTD knew that the novels had pretty well established that the Gallifreyan equivalent of being a teenager is either 70-120 or 90-120.
The 6th Doctor once gave his age as 900. (I think, but cannot be certain, it was in "Revelation of the Daleks", when he was talking to one of the "undertakers" in that story.)
Revived Series Edit
The new series makes things more confusing by having the 9th Doctor's age be 900, when even the 7th Doctor was older than that. However, ignoring that, the Doctor's Wife seems to put him in his 3rd century when he and the TARDIS stole each other, which fits with the Fourth Doctor stories.