There is a nice Betta Disease and Treatment thread in this Forum that is at the top. They have alot of suggestions for certian types of betta illnesses. Here is one about fin rot...which i'm not sure if this is his case or not since the tips seem to have a white fuzzy look to them. It may be fungus, but i'm not really sure. Either way, the AQ salt is a good starting point for treatment.
Tail rot or fin rot
•Symptoms: Betta’s fins and/or tail seem to be getting shorter and shorter or they seem to be falling apart and dissolving, Black or red along the edge of the betta’s fins/tail, Bloody tips, Behavior may not change
Treatment: Conservative: Treat with Aq.Salt at 1 tsp/gal. Increase water changes to 100% daily. Replace accurate amount of salt following water changes. Add Stress Coat to help repair tissue. If there is little to no improvement within the first 5 days, you can increase the salt dosage gradually to 2tsp/gal but do not continue any salt treatments past 10 days. Medication: If Conservative treatment is ineffective use API Tetracycline, API Fungus Cure, API Triple Sulfa, OR API Erythromycin. Also add Stress Coat to help regrowth. Continue until fins/tail stop receding and start showing some new growth.
The thing that worries me the most in the picture of him in the cup is that his fins look clamped, and that could be a sign of illness. I would continue with the daily water changes and add the proper dosing of AQ salt for your tank. If he seems too stressed out though for the daily changes..at least do every other day. As far as eating goes, I know that with the first Betta fish I got, it took him a long time to eat anything because he was trying to adjust and he was stressed out because of the new environment. Offering him a place to hide would probably benefit him alot. Some types of pellets are also sometimes too big for the Betta to swallow. I know that with my betta I have now...he would try to eat the pellets I gave him at first and they were too big for him to swallow...so I got him The Hikari Betta Bio Gold pellets (I think thats what they are called) and he had no problem chowing those down. Sorry for such a long post! I just know how frustrating it is to have a sick fish and not know what to do! Sakura8 has helped me out alot here on this forum, and i'm sure she wouldn't mind you pming her to ask her about your fish as well. I hope he gets better!