Will a 350 watt UPS be fine for my computer?

grab one of those kill-o-watt devices to monitor your electricity usage.
if you find that the readings of watts isnt entirely accurate, and your power supply is going dead faster than it should.
you could always switch over to the amperage portion of measurement and specifications.
the option is watts or amperage hours (or whatever the amperage per ___ is used)
that means if you find some refurbished ones and get the wrong size, you should be able to send it back and get the next bigger size (or two sizes up.. depending on your amperage)

volts * amperage = watts
but if you can get your amperage, you might get a more specific number for your needs.
it depends on the uninteruptable power supply.
when they say it is 350 watts.. the voltage and amperage might be different.

electricity works pretty much like this (but the numbers arent exact)
if you have 1 volt and zero amps.. you have one volt  
if you have 2 volts, that gives you 2 volts and 0 amps ... OR 1 volt and 1 amp

since you dont know what the voltage is really going to be from the UPS .. knowing the amperage amount will probably get you closer to the amount of time before the battery runs out.
using watts could prove to be a guessing game, where you buy a UPS and it might be a little bit more than what you need (and last a little bit longer than advertised)
or
the UPS might be a little bit less than what you need (and the battery will go dead quicker than advertised)

you wouldnt want a UPS that says you have 60 minutes, and you plan on saving your work in 45 - 50 minutes... only to see the battery go dead in 40 minutes.

really record the usage, because sometimes those numbers are for new and fresh products, and those numbers might get higher as they age.
same with a UPS maybe?
if those numbers listed are for new and fresh batteries.. what do those batteries do when they age?
do the specs go down and stay that way for a long time?
and if they do go down.. will it be enough for your computer and monitors?