Is renting a machine hourly for running game servers a common practice?
Yes and no. For companies that are paying salaries of people, the cost of having some hardware always available for testing is small by comparison.
For indies who are self-funding, yes, renting hardware by the hour on Amazon or something similar is not a bad choice, as long as you understand the Amazon platform, and can live with virtualization.
Twitch FPS games may be somewhat sensitive to virtualization jitter, depending on specifics. It sounds like your game wouldn't be, so that's good.
One you're "deployed" you will typically have the machines available 24/7 for players, and if you are planning some marketing or other growth strategies, you'll want some extra machines ready to meet unexpected load.
For a game with FarmVille-style growth (very, very rare) you will have to automate the process of automatically inserting new capacity to the system, and you'd better be on a provider with sufficient spare capacity to soak up all your new users.
Smaller cloud/infrastructure companies typically have an upper limit to how much you can provision in one go, and typically have larger granules of payment than "by the hour" -- many charge by the month.
Similarly, the time to spin up a new server on the largest services is measured in seconds-to-few-minutes, whereas the time to spin up a new server on smaller places may be up to several hours.
So, it sounds like you're in the beginning of this journey. I agree with frob: If you can generate synthetic load/players, then that's great!
Apply that to your existing host, and see how far you get. Run a profiler on the server while you do this, to find what the bottlenecks are, which will likely show you some easy improvement opportunities.
This may be in database access, or in CPU processing, or in memory leaks, or in insufficient caching, or in one of a myriad other places -- you won't know until you measure!
Once that's all done, you should be able to run a beta for friends-and-family-and-friends-of-friends to get validation at small player counts.
After that, feel free to open it to the greater Internet. The biggest challenge at that point is getting anyone to actually care about your game, instead of the 100,000 other games that are out there, trying to get people to care for them :-(