It is possible to cycle any size tank given that you don't do 100% water changes, this ensures that there is just enough ammonia retained to keep your bacteria going. All you need is a bit of ammonia, a porous surface and water flow to get the ammonia to the porous surface. That being said, make sure the water is tested almost obsessively until you're sure the tank is established and keep in mind that the larger tank is always the better option.
My crowntail is having the same sort of thing. He used to sit at the bottom and not move to much. When he was in his 5 gallon he was fine and rather active then I added a bubbler(since my plecos kept dieing since there was no filter or anything) then I added the fliter and he started sitting low but still moved around a bit. Then I moved them to a 10 gallon and he sat at the bottom and wouldn't move around to much. With the addition of 3 silver mollies he seems to be really angry by this so those might go back tomorrow. I dunno about all them together
For a 1 gallon tank, water changes have to be done more frequently. I do a 50% water change once a week for my 6 gallon, and it has the whole set up. I'd do at least a 50% water change every 2-3 days. ): If possible, invest in a bigger tank, because frequent water changes can stress your fish out.
some bettas are just inactive in general I have two males that swim like crazy and another that just sits on the bottom a lot :) I think he should be fine! If you are concerned you could just continue to keep an eye on him and look for signs of illness
First of all, I agree with what is being said - 1 gallon is way toooooo small for a betta. 2-2.5 gallons is definitely recommended. Second, how did you cycle a 1 gallon? Once again as mentioned before - I too have heard it is hard to cycle anything smaller than 5 gallons. And to answer your original question, my betta does lay on the bottom of the tank resting and will randomly dart around. Flair is usually hanging around the bottom or in his cave if no one is in the room.
Oh and I'd be doing more frequent water changes on a 1 gallon tank also.