If I remember correctly, those kits come with one of those expanding peat pellets...
Now that your tree has started to be more than a sprout, you'll want to pot it up into a larger pot. Use a well draining soil mix. Too small a pot will dry out too quickly (like you noticed) and also restrict your trees growth...
Yes, bonsai are in small pots, but they don't start out that way... Only when they are 'finished' do they go into the small pots. First you have to grow he tree large enough for it to develop a thick enough trunk to look like a miniature old tree instead of a seedling (stick in a pot)...
Maybe, technically you could leave it in the small pot, but it would take YEARS (or decades) longer and be way harder to keep alive.
There is another thread rigt now on the age of bonsai, which has a reply you might find useful

A larger pot will provide a more stable environment for your tree roots and shoul help it be happier.
Edit: I just remembered reading somewhere that people living in very arid areas sometimes use larger bonsai pots than is conventional in order to prevent them drying out too much... So really, don't get hung up on the size of pot just yet...