i agree he'd be better off in the 5 gal fish in cycle. set up the tank how you want it(make sure you use decholrinator...pond dechlor if you can find it is more economical...same stuff just concentrated...more value in the long run especially while cycling and doing lots of water changes...... and run the filter for half hour before you acclimate him JUST to make sure things are working properly...be usre to rinse the media first as if you dont get the "dust" off of it may make the tank go cloudy for a day or so.
acclimate him into his new tank as you would if you got him new. net him...put in a container with old tank water and float him in the new tank for 10 min.,...add some new tank water to container..float him for ten min..add some more new tank water...float for ten...then net him and put into the new tank...it is not good practice to add water between tanks. JUST in case.
api kit is perfect!
make yourself a chart with the things you'll be testing on a piece of paper or notebook. date, time, Ammonia reading, nitrite, nitrate and ph. and i always have a miscellaneous column for if i added decor, water change, plants, other fihs etc (like a comment column)...every time you test, WRITE IT DOWN...gives you something useful to look at while your cycling and can ask questions...in the meantime read about cycling and ask questions...might help you out later if things come up ("my betta's acting strange" quite often its a water quality issue...if you ever get other tanks, will be valuable too. you live and learn.
and while cycling test test test (water change water change water change)...he'll be fine! you are doing great for him by getting him a bigger tank!
also think about getting him plants...dont underestimate the power of plants to use up ammonia and other nitrogenous things)... real is best, but if you go fake, get silk...plastic tends to have sharp edges that might snag their long fins. they LOVE to rest on plants...the more the better, really.
best of luck...he'll be a happy chap!