Though I feel like I'm going against my better judgment, I would go with an inexpensive fixed gear. The problem with old road bikes is that they tend to be either poorly taken care of or insanely overpriced (some, sadly, are both). Assuming you can get a decent ten speed for a hundred dollars or so, count on having to put money into a chain, brake pads, tires, possibly a wheel or wheels, maybe a cassette, and any number of other things that can go wrong. Compounding this issue is parts availability. While most the parts are usually locatable, it's not like you can walk into any bike shop and pick up a five speed cassette. If you're unlucky, you could end up with something with strange threading or parts diameters, and that's not fun either. A good friend of mine has a really nice 80s Fuji, and he's put probably $400 in the last two years in maintenance and parts into it in addition to the $100 or so he paid for it. It's a great bike, but it wasn't as cheap as he'd hoped, and things still go wrong with it fairly regularly (one day the wizened headset got loose and never tightened again). To be fair, he would have saved a lot of that money if he learned to work on the thing himself (or had let me do it).

The advantage of a cheap fixed gear is that there's simply less to maintain. There are plenty of rideable options under $400 brand new, and they are easy to work on or find parts for. Flashy isn't a good idea for a lockup bike (so avoid, say, this nightmare, which I promise is even more garish in person), but something simple and inexpensive combined with good locking skills and a little bit of luck and you should be fine. If you can find something used in your area (without rust or Deep Vs that match the chain), even better.