Yes, you could use water from your other tank but it's better to bring over an object, such as peice of bogwood or silk or plastic plant or gravel. I don't know if anachris will help the cycle (it might) but it will help to improve the water quality and give your guy a nice place to hang out You can cycle a 7 gallon with a betta in it, just get a liquid water testing kit and test each day. Initially, you are watching for amonia and doing a partial water change when the amonia goes above .25 ppm. If amonia is at .5 ppm, do a larger partial change immediately.
Can you afford the amonia only liquid testing kit? Petco sells it for around $9 or $10 dollars where I live.