Welcome to the forum and betta-keeping! My number one tip for new betta owners is to get a heater and a thermometer. His water may be warm enough now, but come autumn, the temperature will drop rapidly. Perhaps even before you notice it.
Another good tip is to not use the filter. I know this may seem odd, but it's not going to do anything in that tank aside from move the betta around. Filters are used to provide extra surface area for the beneficial bacteria that comprise the cycle to grow. Your tank is too small to cycle, so the filter has no use.
Welcome to the forum you are in the right place to learn and ask questions about the proper care of Bettas. First off keep his water clean by doing water changes and always use water conditioner at each water change. Heaters might be hard to find for a tank your size but would not cost much and he needs one when the weather cools off, they need a water temp of at least 76 to 78, 78 being the best of the two.
No over feeding to can make his tummy bloated and sick. Betta pellets are a good choice and feed 2 or three in the morning and the same at night and turn the light off at night so he can rest and yes they do sleep and he might sleep in the weeds you say you have. Something he can hide in would be nice for him they enjoy caves and such things but make sure the holes are big enough so he doesnt get trapped in one. Enjoy your Betta I bet he is handsome little guy.
I suggest you get a air pump and bubble stone as an alternative to a filter. This will help circulate the water gently and provide oxygen in the tank water. When doing daily water changes always acclimate your betta by putting him in a cup he probably came in and float it in the tank for about 30 minutes. If you want to save time keep a empty 1 gallon water bottle and use it to store your pre-treated water with water conditioner. Wait 24 hours before using it so it can be room temperature. Just be careful you don't overfeed him. Most people feed 2 or 3 times a day. I feed my betta's no more than 5 pellets a day 3 in the morning and 2 in the evening (Personal preference).
If you notice your betta keeps spitting out the pellets and try to eat them again its normal. It just means the food is to big for his mouth. Then your option is to chop them up or by micro pellets which are easier for your betta to eat.
Thanks :) It has a lid, and as for heaters, are they big or expensive?
I'm assuming it's summer where you are and that the weather is currently warm. If that is the case, you may want to get the 25w adjustable heaters on ebay that ships from Hongkong. If that i not the case, get a 25w adjustable from amazon :)
This may not be an option for you, but I recommend you to get a 2.5gallon tank minimum. Much easier to fit a heater and will probably only need to do 100% water changes once a week. Thats how do it. Not everyone is happy with that though lol.