Professional only unlocks after beating the game at least once on either Recruit or Contractor difficulty. Enemies hit harder and there's much less time for your partner to revive you when you are down.
NOTE: You can go to Mission Select and play each mission on Professional out of order and it will still unlock.
All Primary Weapons unlock after your first playthrough, so save your money for upgrading the firepower of your favorite guns. The Dao-X kills most enemies with 2 shots and also knocks down even heavy armored guards at mid to close range.
Play slow and clear out all enemies before advancing (the majority of the the enemies in every area are finite, so you will eventually kill all of them).
You can unlock this with either the AI or a co-op partner.