Having just done all this, don't get a UPS for the purpose of powering a router.
A UPS is designed to provide a fair amount of power (think a computer) for a few minutes, enough to shut it down cleanly. That's all. It doesn't last long, nor is it designed to. It's also REALLY inefficient, so if you think running a tiny router will make it last longer, it won't make much difference. The efficiency drops hugely as the power demand reduces, so that it's consuming more power running itself than supplying your router.
I'm not sure where you've got your information from - that's not the real world - a UPS can last for as long or as short a time as you wish.
Modern small UPSs have efficiencies in the 90+% range, so they are very efficient - for their SmartUPSs (a very typical small domestic UPS), APC (*) quote figures of up to 98% - see
Efficiency: SUA750IX38
The perceived efficiency figure drops off sharply at low loads only because the energy usage of the UPS itself is pretty constant (charging batteries, supplying the internal electronics), so whilst the output energy falls, the ratio of "lost" to useful energy appears to increase, i.e. the UPS seems less efficient even though the actual conversion efficiency is still very good - modern inverters are extremely efficient at a wide range of loads.
It's much more sensible to look at the real-world run-time for a range of loads for the same UPS - see
Runtime: SUA750IX38. When looking at run-times, you are off utility power therefore looking at real output conversion efficiency only as you are not charging the batteries.
You'll notice that the run-time vs load is inverse linear (as the load decreases, the run-time increases proportionally), i.e. the perceived "efficiency" is not a meaningful statistic in this scenario, i.e. for small loads.
In practice, you can use a small UPS like this to run your core networking for an hour or more, depending obviously on the load and sizing of the UPS - if you have a mix of mains-powered (using IEC/kettle connectors) and low voltage (5/9/12V DC), then a UPS is a good one-stop-shop.
The low-voltage DC-only boxes are good when there's no 230VAC requirement, but even then they'll have internal boost & buck converters to get the correct DC voltages from their battery pack.
(*) I'm just using APC as an example as they are pretty popular and respected, plus I have several around the property, either 750s or 1000s. Any of the decent manufacturers have similar devices with similar characteristics.