In Day of the Doctor, the War Doctor says he is 400 years younger than Eleven in reponse to Eleven saying he's 1,200 years old. This make War 800, or we assume he's rounding to the nearest hundred at most 850, in the episode he regenerates. Nine says he is 900 in Aliens of London. So between his regeneration and his fourth episode there have been at least 50 years for him.
There are two points these 50 years, (presumably including the solo adventures Clive has records of) could have happened: before Rose, or in the gap between him leaving at the end and coming back to tell her the tardis also travels in time. It was a few seconds for her but it's a time machine so could have been years for him.
In Moffat's novelisation of Day of the Doctor, he mentions that Nine on the day of his regeneration smashed all the mirrors in the Tardis because he couldn't bear to look at himself after what he thought he'd done. It does seem like a very casual way to finally decide to see what he looks like now after 50 years though.
It seems most likely to me he's recently regenerated and then has a series of adventures before remembering to tell Rose it's a time machine, probably after something goes horribly wrong (maybe the Titanic or Krakatoa - not being able to stop them in his very own Time Lord Victorious scenario) and he realises he needs someone and nobody he's met measures up to Rose.
That said, I think RTD has said Nine wasn't just regenerated - a quick google found this quote from him: "Well, I hate being prescriptive here, cos sometimes, when I give an opinion on a scene, because I'm the writer and producer, it can become a fact. When I much prefer it if you make your own mind up. But enough time has passed now so, for the record... No, I do not think he'd just regenerated. If you have certain physical features like big ears, or buck teeth, you look at them and sigh at them every time you look in the mirror. And I think, if you'd had eight different faces, even if you'd been in this current form for a hundred years, you'd still mutter at them. So it was meant as a nod to the fact that he'd once had other faces. But I wrote the Titanic stuff and Krakatoa assuming that the Ninth Doctor had been around for a while. He doesn't act very post-regeneration, does he? He appears in command, waving a bomb. This is a man who knows himself, and has known himself for a while." (Source: https://scifi.stackexchange.com/que...n-after-regenerating-did-the-doctor-meet-rose)
So basically the intention is no, he hasn't regenerated. But RTD does say he doesn't want it to become a 'fact' so it's really up to you
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