No.
Look at the UPS as power strip with multiple outlets. Does power strip alone consume electricity or help to save on it? No, it does not.
Line-interactive (and stand-by) topology UPS is similar in operation as normal power strip has it. It takes electricity from one end and provides 6 outlets for other hardware. But what UPS can also do, is when there's blackout, it keeps the connected hardware operational until it's battery capacity allows.
UPS actually increases electricity consumption since compared to power strip; it has battery in it, which it has to charge if you keep your hardware running on battery power too long.
In general;
UPS pros - keeps hardware running when there's blackout (also protects against surges, brownouts and the like due to AVR in it).
UPS cons - consumes more electricity since it's battery needs recharging after using UPS without main power.
But what does matter to the electricity consumption, is the PSU you have. Namely it's efficiency.
Your RM1000x is 80+ Gold efficiency PSU. My Seasonic PRIME TX-650 is 80+ Titanium efficiency PSU.
80+ efficiency chart:
https://en.wikipedia.org/wiki/80_Plus
Your 80+ Gold RMx is 90% efficient on 20% load, 92% efficient on 50% load and 89% efficient on 100% load.
My 80+ Titanium PRIME TX-650 is 94% efficient on 20% load, 96% efficient on 50% load and 94% efficient on 100% load.
To run some numbers;
Let's say you have 300W load on PSU. Your RM1000x would be 90% efficient, meaning that PSU pulls from the wall (or from the UPS) 330W while providing 300W to the components. The extra 30W of power is wasted as excess heat due to the PSU's efficiency.
Now, if you'd have 80+ Titanium PSU, e.g
Seasonic PRIME TX-1000, on 300W load, it would be 94% efficient, whereby PSU pulls from the wall/UPS 318W and providing 300W to components. Wasting only 18W worth of power as excess heat.
So,
80+ Gold 1000W PSU on 300W load, pulls 330W from the wall/UPS.
80+ Titanium 1000W PSU on 300W load, pulls 318W from the wall/UPS.
Difference of 12W per second. Not much, but if you add it up; 120W per 10 seconds, 1200W per 100 seconds etc, it makes a big difference.
80+ Titanium PSUs are the most efficient PSUs out there. Due to that, they also cost a lot of money. But if your goal is to save on electricity, you'd want to use the most efficient PSU possible. Also, more efficient PSU means that you'll get longer runtime out of the UPS as well. Since it makes a difference if PSU pulls 330W or 318W from the UPS.
And this is also one of the reasons why i have 80+ Titanium PSUs in use with my UPSes (to get longer runtime and to waste as little electricity as excess heat as possible).
That being said, you could look towards more efficient PSU. It has good benefits for you situation. Only downside is higher cost of a 80+ Titanium PSU.
In general, to save on electricity usage;
* don't use stand-by/sleep modes on any hardware (TV, PC, consoles etc). Since stand-by/sleep modes still consume electricity.
* don't keep lights on in the rooms you're not in. (E.g my missus has a bad habit keeping lights on in kitchen at all times when it is dark, despite we not being there.)
* use LED light bulbs in light fixtures. LED light bulbs consume fraction of the power, compared to the incandescent light bulb (about 10 times the difference) and to the halogen light bulb (about 6 times the difference).
* buy more efficient appliances (in EU, big home appliances like fridge, washing machine etc, have efficiency rating on them, so you can see which are more energy saving appliances).
* with washing machines and dishwasher, don't run them on quarter of half load. Run them at full load capacity to save on electricity and water. Dishwasher is actually convenience. If you have 1-2 dirty plates, better to manually wash them clean in sink, rather than using dishwasher. Same with washing machine, don't use it to wash clean 1-2 clothing articles. Collect the dirty clothes until you can get a full load. Then wash them. Also, newer washing machines have Eco preset. You can use that if you like, but it takes far longer to complete, than regular program. It can save on water though.
* depending on what electricity plan you have, e.g different rates for daytime and nighttime; electricity is usually cheaper at nighttime. So, better to run your washing machine (and dishwasher) at night, to save on electricity.
* in private home, when you have heat pump for warmth, set the output temperature few degrees lower than you've used to. Sure, it is a bit chilly when room temperature is 21C compared to 24C but that 3C degree difference is a huge electricity cost for heat pump to maintain in entire house.
* when you have water boiler for hot water (no central hot water as it is usually in apartment buildings), taking shower consumes 2-3 times less water than taking a bath. And since hot water is heated by water boiler, water boiler also takes a lot of electricity to heat water. When taking a shower, don't keep the water running, since it wastes hot water and electricity needlessly. Instead, get yourself wet, close the water, soap up, scrub and once you're ready to rinse yourself, then turn the water on. It saves on water and electricity too when you have water boiler. Actually same is when washing hands. Don't keep water running at all times. Wet your hands, turn off water, soap the hands, scrub and for rinsing, turn water on.
There are other electricity saving costs as well, but those should suffice.
Oh, don't know about your situation, but i can choose between 5-6 different electricity providers, from whom i get my electricity. So, when one provider's plan is too expensive, i can change the provider who offers cheaper price per kW/h. And within each provider, there are 3 different plans i can pick. Fixed price for kW/h, Exchange price for kW/h and Seasonal price for kW/h (seasonal plan has exchange prices for summer time and fixed prices for winter time).
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