My opinion is any automatic in your price range is problematic. There were several different automatics used in various engine/year combinations, but all have their problems. Not saying there are not thousands of them on the road, but they can and do cause trouble. The repairs run from mildly expensive to more than your budget for the entire car, depending on which one. If it is at all possible, teach her to drive an manual. The 5 & 6 speeds are very reliable.
Having said all that I have a daughter that has driven an ALH (TDI) Golf with the 01M automatic since 2006. Already replaced it once, the second one is still going strong, but they can and do break without warning and they are expensive to repair. If you buy one, I would expect problems with the auto trans at some point.
I can't speak to the gas engines, but the TDI's are very nice engines, especially the ALH engine. If she will be driving long distances, it may be a good choice. They will easily run 300,000 miles or more but you must maintain them. You cannot put off servicing - this goes double for the timing belt - or they will cause nothing but problems. If you are considering a TDI, I would read as much as you can at TDIClub Forums - Powered by vBulletin before you buy.
Whatever you decide, I would hold back some money for general repairs. It's likely in this price range it will need freshening up. If you can do the repairs, I would look around for one that needs some work. You can pick them up for cheap. They may need a couple thousand in repairs, but you know what was done and once you get it sorted out it should be good to go.
I bought a 2000 TDI earlier this year. It needed a ton of work, but now is my wife's daily driver. I wouldn't hesitate to drive it anywhere and the 40+ MPG is nice.