Hello @gustavjung! Apologies for the delayed response here. I just wanted to say that you're not alone - unfortunately, I think there is often a trade-off between price and ease-of-set-up.
Generally speaking, if the hourly price for cloud vendors (you could also take a look at DigitalOcean, Azure, etc.) is prohibitive, then you are definitely leaning towards running your own server. This comes with some level of linux admin experience necessary (and in fact, so do the cloud vendors).
If that is where you are forced to be, I would take heart in the fact that it is learnable and do-able, and there are very good resources out there on the web for linux administration! Also, it is a great skill-set to learn
One other option - it is a bit more advanced in one sense, but for ease of installation, docker or (more specifically) the rocker project maybe be decent options for installing RStudio Server Open Source and Shiny Server Open Source onto a linux server. It adds a layer of virtualization that can be confusing / tricky to get right, but docker is a useful tool for software installations and can work very nicely if you set up volumes / persistence properly!
Worth a gander at least I hope it's going well!! Hang in there!