Can I use a programmable thermostat with an oil furnace that circulates water?

It will only work if you have a zoned system which, it appears (since you only have the one thermostat), you do not.

Not enough information to hazard a guess as to how much it would cost you to create a zoned system, but certainly much more than you'd want to spend on a rental unit.

Cheaper and MAYBE workable solution would be to get a plumber in there to see if the system can be balanced out so you're putting more heat into the downstairs rooms and less in the upstairs and provide the bulk of the heat upstairs via convection.

In that scenario, I would think that you could go ahead and install a programmable t-stat but be advised it's not going to change your situation by itself. All it will do is ensure that the downstairs freezes and the upstairs is hotter than hell at specific times.

You can't put a replacement thermostat where there is no thermostat to begin with. And if upstairs is hot, downstairs is cold, a programmable thermostat isn't going to solve it.

Programmable will just automatically change the set temp at the set times.

Get some box or window fans and set them up to blow air through the radiators on the first floor where it is too cold. You can even buy fans with built in thermostats so the fan will go on only when the radiator is hot.