x2 to the comments about electrical safety. Household current is designed to cause muscles to spasm (that's why household AC at 120V alternates at 60Hz, and household AC in Europe alternates at 50Hz - those values cause human muscles to spasm, which will often break the circuit fast enough to save lives - DC at 12V does not.)
That aside:
I have a generator as backup for the tank, but dry camp and boondock a lot (that's to say, camping in a trailer without electric or other hookups) so have some experience at figuring out how long things can operate off of battery alone, or battery-backed solar.
A 200 watt heater pulls roughly 1.67 amps at 120V, so will pull roughly 16.7 amps at 12v, plus losses from whatever inverter you use to convert 12V DC to 120V AC. Figure it at 18 amps.
When looking at batteries for this application, the only number that matters is the reserve capacity. That's a measure of the number of hours the battery can provide an amp of current. The Optima YellowTop D51 has a reserve capacity of 38 amp hours, which means it can provide either 38 amps for an hour, or 1 amp for 38 hours. It can cause premature failure of the battery of it's discharged below half the reserve capacity repeatedly, yielding a useful reserve capacity of just 19 amp hours.
In other words, it will keep your heater going for just slightly over an hour (or a bit over two hours if you don't mind destroying the battery). That's probably not really long enough to be useful.
The best bang for your buck in terms of deep cycle batteries generally comes from 6V golf cart wet cells (GC6 - a pair wired in series will produce 12V and can be bought for around $200). Those will provide 220 amp hours, or a useful capacity of 110 amp hours which would run your heater for about 6 hours - probably enough for overnight (keep in mind, your apartment is getting cold, so the heater will run more than normal). The problem with these, and any other wet cell, is that they discharge hydrogen while charging, so can't be used indoors. For considerably more money (think $1000), you can get AGM versions that are safe for indoors (UPS systems use AGM batteries)...
Another problem you're going to encounter is a need for a way to QUICKLY recharge them. Car alternators typically charge at 50-70 amps, so a car would take about an hour (or less) to recharge the D51 Optima, and about 3 hours to recharge the partially discharged wet cells described above. In either case, that's not very efficient and likely leads to other problems (running out of gas) if you're snowed in... Also, the wet cells above weigh about 75 lbs each, making transporting two of them back and forth between your apartment and the car a non-trivial task.
So what can you do? Not a lot ... If you have the space (think: 10 feet wide, 5 feet tall), a pair of 200 watt solar panels will provide enough power to run the heater while the sun is shining, plus a short time after it goes down (hooked up with the Optima D51)... Probably not enough... Also, they're heavy (~45 lbs each, if you have to move them around, and fragile) and expensive.
My suggestion is to seriously consider how you can put a generator on a balcony, or locked to a tree outside the apartment, or something like that. An inverter generator like the Honda EU1000 is small, lightweight, relatively quiet, and wouldn't use a lot of gas.
