As Matti-patti said (and the quest giver does as well). you don't get the benefit until you practice it "when you really need it" (if I remember the NPC's wording correctly). In practice, it seems to mean using the skill against a live target, such as a trapped bird, is sufficient. It can also be noted that this skill increase is not part of the standard training system, so you don't get any "skill increased" indication (but if you check the number before the bird assault and after you'll find that the number has been increased).
I don't know what happens if you have an outstanding skill reward for a skill and get another one for exactly the same skill, though.
Thus, you'll have to get a sword to be able to get the reward. However, I haven't heard of any indication of it expiring, so you can take your time (I believe it took my latest character a few months to get a sword to get exactly this reward, and it worked).