First let's get clear on terms. A "UPS" is an emergency battery that is designed to provide 20 minutes of power, enough to do an orderly shutdown of services, database, and operating system... and flush the caches of DASD (hard drives). UPS's are sized for that task, with a very low battery:load ratio since it's only sized to run 20 minutes.
What you're looking for is essentially off-grid power, which is sized to provide 24x7 service while intermittently replenished from one source or another. Much larger battery:load ratio.
Your "5 volt 1 amp" requirement is the same as a standard USB charger - the cheap/common ones, not the better ones intended for iPads.
Kill unnecessary power conversion!
Your load is a piece of low power electronics. Most actually run on low-voltage DC. That's what is going on with the really weird power cord with the thin wire and a big lump on the wall socket end.
Batteries are the keystone technology that you can't do this job without. And by amazing coincidence, batteries are low-voltage DC!
So this is a gigantic opportunity for us. If we can stay in the low-voltage DC realm, that means we don't need an "inverter" to make 120VAC... and we don't need a "wall wart power supply" to convert 120VAC back to low voltage, in a pointless round-trip.
Why does that matter? Because power conversion is not 100% efficient. It costs us coming and going - and it costs us in energy. That means a bigger battery would be needed to power the conversion losses alone. On a very small load like yours, the parasitic loads could be 3/4 of the total load! A surprising number of people don't care and are happy to rack more batteries for the simplicity of a double conversion. So your call.
If you use common/cheap 12 volt batteries, you may need a single conversion to get from 12 volts to 5 volts. Fortunately, such products are easier to find in stores than eggs, at every gas station, liquor store, cell phone store, box store, dollar store, you name it. Right at the "end cap" no less. You've seen them. They plug into a "cigarette lighter" socket.
That, with a 12 volt battery, you're all set.
Could a USB power bank work?
Your load has a nameplate of 5 volts 1 amp, or 5 VA or we can call it 5 watts. To run this load 12 hours, you'll need 5 x 12 = 60 watt-hours or 60,000 miliwatt-hours.
If we're looking at amp-hours at 5 volts, then 12 hours needs 12 amp-hours or 12,000 milli-amp-hours. Well, heck, You can get USB power banks that size or larger! (larger is always good). Problem solved! You just need to make sure the bank can supply a load while it is charging, and is able to continue to do so when not charging.
Otherwise, derating needed.
Unfortunately we need to start derating that 60 watt-hour figure. If we're going to use a 12 volt battery and a drugstore USB power supply to convert to 5V, then we need to account for maybe 20% voltage drop. So now we're at 72 watt-hours.
If we're using a lead-acid battery, and we want long life out of that battery, we don't dare plan to draw it down by more than 25-30 percent. (because after best laid plans go a little awry, we'll be at closer to 30-40% which is the absolute maximum you dare). So that derated off our 72 watt-hours means we need 200-300 watt-hours of lead-acid.
Considering lead-acid
A lead-acid is nominally 12 volts, so 200-300 watt-hours happens at 16-25 amp-hours. Conventional logic is "you should buy the smallest lead-acid that will do the job because that will be cheaper". Most people follow that principle blindly, without actually checking whether it will be cheaper lol. However there is a "pricing sweet-spot" at a couple of points. First, automotive/trolling deep-cycle batteries, which typically sell for ~$100 for ~1000 watt-hours of nameplate capacity. Second is "used golf cart batteries". A golf cart battery is considered "failed" when it can't make a full 18-hole round of golf. They are 6 volts so you need two, but still, far cheaper than even $100 and probably your cheapest option overall.
And these can be acquired locally, so you're not being crushed by shipping costs (lead batteries are heavy as hell) - and guess what, when you buy with Prime, the seller has padded the price to account for that.
What I am viewing (on amazon) is something like 600VA for $70 lead-acid but I do not see any kind of amp hour rating.
Don't buy stuff like this on Amazon because a) you'll get mauled on markups due to shipping lead, and also b) much of it is poorly made or outright counterfeit garbage, and that's more likely when the seller does not disclose specs.
Considering other chemistries
Some other chemistries may let you get to 5 volts directly. If you can produce 5 volts directly without any conversion, then conversion losses go away.
Nickel-cadmium and nickel-metal-hydride are 1.25 volt cells, so 4 cells in series are 5 volts nominal (higher at first, less later). A NiMH "D" cell is 10 amp-hours, so that is about 12.5 watt-hours. Four of them in series is 10Ah @ 5 volts, or 50 watt-hours. That might suffice since I doubt your load is fully 1 amp all the time. If not, series-parallel two strings of cells. Nickel batteries can work their entire range.
Lithium family batteries produce 3 to 4 volts nominal, so either the device needs to be happy with that, or you need to stack 2 in series and use a voltage converter (conversion loss) to knock it down to 5 volts. Lithium batteries can reliably use about 80% of their stated capacity. The basic 18650 lithium is 3.7V for 2500 mAH, so about 9 watt-hours per cell.
Any of these solutions will require a charging system that is designed to match the battery chemistry involved - right down the the particular sub-type of lithium. I advise using lithium batteries with protection circuits built in.
