Welcome to the forums, and congrats on your first betta.
Don't stress too much, as a lot of people don't fully understand what is needed to properly care for a betta. Too many myths and pet store employees/people not doing the proper research into these fish.
The 1 gallon is a large enough tank, as Oldfishlady mentioned- weekly water changes of 1 50% and 1 100% is needed, along with water conditioner.
The sticker thermometers can be a little unreliable at times. I suggest an in tank glass thermometer, they tend to be under $2, and will be useful when doing water changes to make sure the new water is roughly the appropriate temp.
What is the name of the live plant that you have? Some require different needs, such as substrate, fertilizer, lighting is always a must, heat..
The ideal way to acclimate them to new water (during the 100% as well) is not only just floating the cup in the tank, but during that time you would want to remove some of the water from the cup (throw it out) and replace with water from the tank a few times in a span of 20 minutes or so.. that way she won't go into shock from an immediate change in chemistry.
You don't need a larger tank if you are willing to do the water changes, and you don't need a filter in anything smaller then 5 gallons, imo, when willing to do the appropriate water changes.
If you ever do decide to upgrade to something a little bigger you can always look at Kritter Keepers at PetCo- normally the larges are roughly $10 for 2.75 gallons. They are heater safe.
I would save up the money for a heater.. look around, you can usually find a 10 watt for roughly $12, which would work good in your 1 gallon.
No one thinks you are dumb.. please don't think that. We just want what is best for you, and your little girl. You are trying to give her the best, and that is a good thing.
Is she still acting lethargic? Or are there new signs?
If so, are you using water conditioner, and what is the current temp of the tank?
*Just realized you are in Sweden.. so I didn't convert the liquid measurements, money.. and unsure what is available to you- store wise..