So don't host it on a cloud provider. You can host it from home, or somewhere else. For example I rent a cheap dedicated server with a 1TB hard drive and unmetered 1Gb bandwidth, for £8 a month, which would work great for something like this and there is no chance of unexpected costs.
https://www.serverhunter.com/ and https://en.metadedi.net/ are useful resources for that. Though search for more information before signing up for anything expensive and/or long term as there are some truly crap hosts out there. There are some pretty good budget hosts in the mix though.
If you can wait until the end of the month you might find black friday offers from some providers. They usually get discussed in places like https://www.lowendtalk.com/ (I'm not sure how quickly such short term offers are reflected on the above server search sites, if at all). Again, do research on the hosts you chose lest you buy a lemon.
OVH's kimsufi series (https://www.kimsufi.com/) has very cheap dedicated servers. I have a KS-2 and am quite happy with it. The servers are very low-end and have old hardware, though. Not a problem for me but you should be aware of it.
I think Hetzner's Server auction (https://www.hetzner.com/sb) has better offers if you can wait a bit for a good deal and you're looking for a little bit more performance. A while ago I saw an i7 4770 with 32 GB RAM and 2x 2TB HDD for ~26€. Their servers also come with an unmetered 1 GBit connection instead of just 100 MBit.
