Hi, I know there are a few tools floating about the internet that allow games to be downloaded or backed-up.
Including the official GOG Downloader.

But what I am looking for is a utility that could sync my games, for instance, daily, and automatically upload them to cloud storage. Is there anything like that currently available? Be it an online service, a utility or even a script that I could run on a server?

Something that would automatically "push" games to cloud without downloading them to my computer first?

If not, anything that I could set up on a linux server that could serve similar purpose?

Well Gamesave Manager has options like that.. although I can't really figure out what exactly you're looking for.

Sorry, I am not looking to back my saves up. Just the installers for the games.
Like what GOG.com Downloader does, but directly to cloud storage, instead of my hard drive.
Something that could go online daily and see if there are new games to be synced.

I used to do that manually every so often, but I buy too many games! And it became a huge hassle trying to figure out what I already backed up and what I didn't yet :D

Hmmm GoGrepo and rsync would probably work.
Or just GoGrepo to the cloud client folder.
I suppose it depends on the cloud provider and the access options it has.

Buy an external HDD - you can pick up smaller size ones for next to nothing (far less than monthly cost of cloud storage), then save your installers there. Clound may have its benefits but big drawbacks is 1) you have no control, 2) without net you have no access. These are also reasons to buy DRM free as the files are yours to use without any requirements.
As you will have access to your GOG account for as long as they exist, that is also a backup - same with Steam though your still tied to the internet connection requirement there.

There's no out-of-the-box solution that I know of.
The Python based GOGRepo script is able to sync your whole GOG library, but you'll have to find a way to direct it to your cloud storage, for example by mounting it to a local drive or folder name to be used by GOGRepo. The GOGRepo thread might be helpful to give you some details on how to use it and what to change in your case.

It depends:

1: If you have shell access to the server and Python is available, you can use GOGRepo there just as you would at your own machine.
2: If you do not have shell access, it depends:
2.1. If you can mount the storage locally (not a syncing solution like Google Drive or DropBox, but for example SFTP or WebDAV is available so you can access it directly), you can mount that, point GOGRepo at the mount point, and run as normal. (Technically you are downloading the data only to directly upload it to the storage solution, so it uses your own download and upload bandwidth and counts towards any possible transfer limit but it doesn't take up much space on your own local storage.)
2.2. If you cannot mount the storage locally, no, you'd need to grab the files before you can shove them up to the server.

Brilliant, thank you for all the replies, I'll go with GOGRepo synced to a mounted cloud instance. This sounds like the best option. :)