When you install it, do not plug the monitor into it. That'll save you some run time in that 2 hour window.
You're getting into expensive territory for 2 hours. Probably outside of your cost envelope.
Assuming the PC consumes 200 watts on average through the night (a low estimate, IMO), You need a stored capacity of 3.4 amps you can draw to run it for two hours.
(200 watts, for 1 hour, at 120vAC, is a draw of 1.66 amp-hours. You need twice that for two hours.)
APC units are given model numbers in "VA" nomenclature. Volt-Amps, supposedly, but they are really optimistic with their estimates. An 800VA model supposedly has 800 volt-amps. That comes out to an ability to run 6 amps @ 120vAC. In theory, that gives your computer (@ 200watts) your 2 hour window.
I'd suggest a SmartUPS 1500VA model of some kind. I wouldn't trust an 800VA model to stay up for 2 hours.
One other concern:
If you lose power and this PC stays on... does your ethernet switch stay on? How about your router? Your ISP's gear out in the street that gets you to the innernetz? You'll need to keep all points of communication up for this interface to run.
http://www.apc.com/tools/ups_selector/index.cfm