You could use a very low power computer that's always on like a Raspberry PI Zero W to send the WoL packet to the backup computer. It only uses about 1 watt. Some routers have the ability to send a WoL packet as well.

I do backups with a Raspberry Pi with a 1TB SD card and leave it on all the time. The power draw is very small and I think reasonable for the value of offsite backups.

My personal experience with WOL (or anything related to power state of computers) is that it’s not reliable enough for something offsite. If you can set something up that’s stable, awesome, but if your backup server is down and you need to travel to it, that suuuucks.

If timers on the bios aren't an option, look for the settings to power on after power outage. If you turn it on, you can use a standard timer plug to turn the power on and off.

Can you set the BIOS to wake up at a certain time? Many have that option.

I use Upsnap on a low-power SBC behind a reverse proxy: https://github.com/seriousm4x/UpSnap

Before that I used the WoL feature on my Asus router.

Home Assistant can send wake on lan commands.

If you have a remotely accessible Home Assistant Server on your LAN it can send the commands for you on the LAN while you access it remotely.

it ain't pretty, and may not quite suit what you're looking for. other comments recommending a pi or similar are likely better taste. but here's what I've done:

I have a cheap Android tablet that stays home and is connected to our home network. if needed, I connect to the tablet with TeamViewer and use a WoL app to send the packet to my computer on that same network.

you can configure bios for self power on at certain time and then turn the server off after finishing backup

Wake on LAN is a LAN feature, not WAN, so you'd need to issue that over the local LAN there at the house. You're going to have a hard time trying to get that working over the WAN (if that's even possible).

The other comments mentioning a scheduled boot would be a much easier/simple solution if it works for you.

But I'll throw this in, the super basic least tech solution to this is to open a port forward to the house's network router. Yes, I know you don't want to do that, but it's probably the only network device at that house that's actually on 24/7 right? And by all means lock it down however you like. My simple method is to open the router login to a non-standard port number, with a IP whitelist, add my own home IP address to that IP whitelist, and bam you now have access to that remote home's router for just your IP address. Log in remotely, issue a wake on LAN via the router's own web ui, done.

It's perfectly reasonable to make this a bit more secure if you wanted but it gets slightly more complicated - open a non-standard port for SSH access to the remote router's SSH port that only allows SSH login with key. Generate a SSH key and share that key with yourself, then you can log in remotely to that remote house via non-standard SSH port using the SSH key (no user/passwords). From there you'd have to see if you can issue Wake on LAN on the SSH command line, or set up a SSH tunnel from that remote LAN to yours so you can proxy into the router login page and do your Wake on LAN from there. ... yes I realize this got complicated :/ But you've got a few things to explore given your patience for tinkering with this stuff :)

Of course much of this relies on that house's router having any of these features to enable and configure. The main takeaway here is that Wake on LAN requires something on 24/7 at that remote LAN for you to enable remote access into and issue a Wake on LAN command within that LAN. How to actually accomplish that is the tricky bit.

Just ask Ian to start your computer when he goes on it