Unless you can find someone here to do it for you, I don't think you're going to be able to avoid learning how to carry out an upgrade.
You can upgrade from 2.12 to 3 and then go to 4, but as you are aware, a fresh install of 4 would certainly be nicer.
I would suggest you install a copy of 4.40 (or 4.41 if it is around by then), and then backup your 2 site, run an upgrade on it to 3 and see if it works. If it does then carry on to 4 (asp.net 2.0!) and if it doesn't then restore the backup and look at transferring your users and content instead.
If you get it up to 4 and it's largely working, then you should still probably export your content for import into the fresh install of 4. If the content is largely in core modules then you'll likely find it remains intact. It is less advisable to export from an earlier version for import into a newer.
Then you only have to worry about exporting and importing the userbase. I've personally used a module off Snowcovered for this and if you get one, then make sure it can export users from 2.12.
However, I note that your site is hidden to casual visitors. If this is not really essential and if there is no chat forum or anything that specifically requires a user account then you could of course consider dropping the accounts and just open the site up. That would save you a lot of drama and you could just get people to make a fresh account when they next show up.
I dont' recall if DNN2.12 had an export-content feature, but if it did, then you could at a pinch simply try it out and see if it will work.
If you are having trouble using the remote desktop to carry everything out manually (I would), then you might find it easier to get a shared hosted account somewhere like Webhost4life, which has a good control panel that makes it a lot easier to create, backup and restore databases. aAbasic account is dead cheap.
I'm afraid I can't really help any further than that. Good luck with the site!
Rob