What are the dimensions of the tank?
Sand substrate looks really nice, I highly recommend CaribSea Super Natural Crystal River. I got a 10 pound bag for $5
For filter, I would think a small sponge filter would work well. My mom's betta messes around in the bubbles and seems to really like them, as long as it does not run too fast. You would need a small air pump, some airline tubing, and the sponge filter. On Amazon you can get a small Tetra air pump for just $5
I don't unfortunately have any recommendations for a heater.
I agree about a nice little desk lamp working well!
For plants, some nice plants are Amazon Sword, Anubias, Vals, Dwarf Saggitaria, Brazilian Pennywort, and Java Fern. Amazon Swords need root tabs, Anubias can't have its rhizome buried, java fern must be attached to some hardscape not buried or it will rot and die.
I highly recommend YourFishStuff for buying your plants, I have ordered from them 3 times and all plants arrived quickly, in perfect health, and they were very affordable. They have a website and an ebay store, I would highly recommend checking prices on both because they do different sales and prices and you can save several dollars. The plants they sent were massive
I would highly recommend adding a piece of dragon stone or driftwood to your tank, it really makes your tank look more natural!
Sand $5
Pump $5
Airline Tubing $2 (I would look at Walmart if I were you)
Sponge Filter $5
Amazon Sword $5 (mine from YFS came with 3 individual plants! They label theirs Echinodorus Bleheri)
Anubias $6
Dwarf saggitaria $6 for 2 or 3. I would not recommend those from YFS
Now about stocking. I personally feel like between 5 and 10 gallons is BEST for a betta, but a 4 gallon is definitely ok if you provide for the bettas needs (warmth, water quality, mental stimulation, and decent space to swim) . I ask about the tank dimensions just so we can get an idea of how much swimming space he would have. If your betta doesn't make it, I would do shrimp and nerite snails.
What is wrong with the betta that he may not live? My recommendation would be to do daily water changes until you can move him over to the 4 gallon, make sure to dechlorinate any new water. I am assuming he is in something smaller than the 4 gallon right now?