If you really need to worry about a sump pump failure, it might be worth making a real investment. For about $2000 you can back up all the main items in your home with a standby generator and automatic transfer switch.
Something like this:
http://www.electricgeneratorsdirect....FQctaQodZtsBRA
I know Honda supports most transfer switches hooking to their auto start portables too, but you'll spend a hefty amount on a new Honda generator.
As for the battery backup idea, look at the watts your pump is drawing and what the battery will support.
Edit (Eplanation):
A 750 VA battery will yield about 60% in watts or 450 watts
A 4.9 Amp sump pump running at 120 V will yield 588 watts
That APC won't power your pump. You need at least 1000 VA in that case. I believe there are some UPS systems in the $150 range with 1000 VA.