Okay first off are the white spots fuzzy or are they just white scales? Keep a close eye on them because fuzzy white spots indicate bacterial infection. If they look like grains of salt or sugar, you have got a case of ich on your hands, a parasite invited by cold water temperatures. If they jsut look like white scales you have nothing to worry about.
While we are on the subject of bowls I strongly advise that you invest in a bigger tank in order to not only ensure that your betta lives a full and healthy live but also so that fishkeeping does not become a chore for you. The average betta bowl (assume that would be 1g) must be cleaned every 1-2 days in order to prevent the toxic wastes your fish produces from building up to lethal levels. A high-ammonia environment will ultimately lead to a premature death and will also facilitate various illnesses such as bacterial infections, ammonia burns, fin rot and parasites.
Furthermore bettas are tropical fish and actually prefer warmer water than most 'aquarium fish' that require a heater. To achieve the healthiest environment for your betta you are looking at about 78-82 degrees, with 80 being preferable. Maintaining such a temperature can be difficult without a heater, most of which will overheat a volume of water as small as 1 gallon.
I recommend you upgrade to a 5g while boxing-day sales are still on. Bettas can live up to 7 years given proper care. For more infor, check out the betta caresheet pinned in the betta care section.