What's your technology stack?
If you have a Python or Node.js or similar there are a wealth of inexpensive options. I personally really like Fly because the pricing is predictable (you effectively pay per container per month) and it's easy to get anything that runs in Docker to run there. https://fly.io/docs/about/pricing/ starts at $1.94/month for a 256MB container which may be enough if your backend is small and efficient.
Heroku is a bit more expensive but still a relatively cheap and very robust and proven option.
If you want to maximize the performance you can get for your money a dedicated server from Hetzner is great value, but you'll have to do a lot more management than you would with a PaaS like Fly or Heroku.
Don't spend too much time worrying about serving your own models. This is a MUCH harder problem - you need GPU allocations etc - and a whole lot more expensive. Fine-tuning models is rarely the right solution so I don't think it's worth assuming you'll be doing that. If you DO do that you can solve that hosting problem separately later on.
(Fly have a GPU product which is great but like all GPUs it's expensive compared to regular hosting.)