UPS’s are fine for short outages and to protect certain equipment from power spikes and surges.
Personally I have multiple small UPSs in the house. They protect my desktop computers, expensive TV’s and so on. One supplies power to he on-demand propane water heater. It needs 120 volts AC to power the igniter and if you are taking a shower and the power fails the water turns cold instantly! Thus, in the off chance, I have elctric for that igniter and can still enjoy hot water.
Don’t expect a UPS to power things for long. It will depend upon how much power the items plugged in require. I have a heavy duty printer and computer on a UPS. If power fails when I am home, I will power down the printer immedietly and start to finish up work on the computer and manually shut that down later.
With Internet you would need to keep power on for both the modem and the separate router, if you have one. You might get 1/2 hour or less time depending upon your computer draw. Also, you might consider an LED bulb on that same circuit to reduce power draw.
I have lately purchased UPSs at Costco in the $100 range. The internal batteries will fail after a couple years. Those batteries are fairly easy to replace, even for a neophyte. The batteries are generally the same ones used for electric gate openers and I can buy them at Home Depot. I also have bought a pack of them on Amazon at a tremendous savings.
Another drawback to power outages and Internet is this: If your neighborhood is out of power, the amplifiers for Internet may be out also. Here in our rural area, someone will take out a telephone pole and Internet, Phone and power all go out (your mileage may differ). Because of logistics here, a single pole out may mean hundreds of us are without those utilities for 6 hours or more. The replacement poles are kept an hour or two drive away. Then they have to dig in rock to set a new pole. Fun Times!
If you are only concerned about your inhouse connections like backing up data from a computer to a local NAS network connected disk drive, or you have a local drive with video/audio and want to keep that running, then you may be fine. I personally have lots of content on a locally connected computer that I can stream from.
A side note. When I worked at an office there was a personal UPS under the secretarys desk. She complained that it didn’t keep anything running during a power failure in the building. I discovered why. Like many UPSs it had two sets of outlest, some were on the UPS itself and supplied emergency power while the others just supplied surge protection. Her computer and monitor were on the surge protection side but on the UPS side and during a power outage, she would be able to sharpen all the pencils she wanted!