Why is NH3 a nucleophile in NH3+CH3 -> H3NCH3
When working with nucleophiles and electrophiles, it is easier to think of them as "positive-ophiles" and "negative-ophiles" respectively.

With that said, NH3 (ammonia) has a lone pair of electrons that is attracted to positive charges.  The carbon atom with three bonds is electron deficient and positively charged (which your image does not show).  Thus, as a "positive-ophile"/nucleophile, the ammonia through its lone pair is attracted to and forms a bond with the positively charged CH3+ ion.

Yes, nitrogen does have a full octet, but its lone pair of electrons is nonetheless still attracted to the electron deficient and positively charged CH3+ ion.  This is similar to how water molecules (also fully satisfied octets) can bind with positively charged sodium ions in solvation process of salt.