I have a betta in a 5 gallon. I did not cycle it first but keep on top of the water changes and water params. for ammonia, nitrite, nitrate, ph. Since you won't have a big load of waste, food, and poop with one fish in that size tank, you should be fine.
Sounds like you already have experience in that field by your signature.
I use an Aqua Clear 20 on my 5 gal. - love love love it. You can adjust the tube to max or minimum flow, it's completely silent, and it's easy to maintain and clean.
Good luck and let us know when you get your new buddy.