Crowntails need deeper tanks to prevent their fins from curling, but your tank looks ok enough. It's best if you don't get a filter. You just need to change the water regularly (50% change) and siphon out all the dirty stuff. Once in a while to a 100% water-change and clean the sides of the wall with a brand-new sponge to clean of all the muck and stuff. You do not need to invest a lot in the betta if you're not going to be be grooming it for competitions and stuff. The only things you'll have to get is
1) Food - You can get Atison's Betta Pro pellets (about $13 per bottle) feed your betta twice a day, see how much your betta can eat then scoop out the uneaten pellets and dispose. bettas won't over-eat like goldfish, they're smarter. it's also good to NOT feed your betta once a week. They're pretty hardy so they can go hungry for 1 or 2 days.
2) Anti-chlorine (you have to use aged water - at least 24 hours old - and anti-chlorine)
3) Aquarium salt (add a small pinch each time you do a water change. it can act as an anti-fungal medium.)
4) A small SOFT net to scoop up the betta when doing 100% water change.
5) A small airline tubing (about 2 metres long) to siphon out the poo.
6) A mirror to be placed beside the tank for 15 minutes a day to let it flare at itself to keep its fins from curling.
7) Cover for your tank to prevent it from jumping (probably an acrylic sheet but please holes to let air in or your betta will suffocate).
8) (optional) a light so that your betta can see its food at night.
9) should you find your betta sick, you may wanna try this remedy called Yellow Powder.
Any other questions for bettas you can go arofanatics forum and ask.