Plastic Surgery for Pointy, Puffy Nipples in Men - Possible?

There are a lot of factors to consider when it comes to puffy nipples. An examination would have to been done to find out what best fits you.
This is a common procedure that most people don’t even know is an option. Surgeons can improve puffy nipples with a wide variety of surgical and nonsurgical simple office procedures under local anesthesia.
This can typically be done in the office under local anesthesia. Usually there are no drains, no outside stitches, and patients can go out in public the same day. Most of the initial swelling is from the numbing medicine and your body absorbs it within the first 4-6 hours. Obviously the body heals with swelling and inflammation and usually that maxes out at two days and then starts getting better. Most of these patients take no pain medicine and usually feel very comfortable going out in public within a few days.  I usually call my patients the night of surgery and check on them the next day. If everything is fine the next day, I then let them shower and they do not need a dressing over their incisions. They can usually start using make up after a few days.  At the two week check up, I usually have people start working out and then after six weeks they can do whatever they want. 
Additionally with recovery, there're many things we do to expedite healing such as nerve blocks during surgery, oral supplements, skin care and scar therapy recommendations, lymphatic massage sessions that are included in our pricing.
I strongly feel that it's very important to have an honest conversation with your surgeon about your particular situation, lifestyle, work environment and specifically which procedures you're having done so that you both have the most honest answer about your recovery and are realistic with expectations and results.
I would recommend that you see a board-certified plastic surgeon in your area that is a member of both ASPS & ASAPS and performs a lot of breast and nipple surgery.
I hope this helps and wish you all the best in your surgery and event. If it does, please take a moment to up vote this answer on the “respond” button on the top right.  

"Puffy Nipple Areola Complex" is among the most common forms of gynecomastia. This gland and/or adipose tissue accumulation can be located under the areola or can be slightly extended outside the areola, causing the areola to appear dome shaped.This form of #Gynecomastia can be found in all age groups but is more common in young adults.As each case is different, it is vital that you consult with a board certified plastic surgeon who is experienced in male breast surgery. The surgeon should determine if you are a suitable candidate for male breast reduction by liposuction or glandular excision. It is also crucial to have a plastic surgeon who is well-versed in the gynecomastia condition can determine if there is firm breast tissue beneath the areola that is causing the nipples to project (which is often the case), and if so, how much of this glandular tissue can be removed to create a flat appearance.#PuffyNipple surgery generally takes only an hour or two, and the recovery time is short; sometimes less than a week. There is minimal scarring and the success rate is excellent. Therefore, patients who suffer from "Puffy Nipples", and who view it as an unaesthetic and unpleasant condition, may desire surgical intervention to correct the "puffy nipple" condition. Men who suffer from large #areolas may notice that after breast surgery, the skin of the chest muscles will tighten, causing the areolas to tighten and diminish in size. This is a result of the removal of glandular and/or adipose tissue.

Puffy nipples can be reduced by removing extra breast tissue directly below the areola and nipple. It is oftne done under local anesthesia as an office procedure. Addtional breast tissue fullness outside of the nipple area is usally removed with liposuction.

Everyone has some breast tissue development to varying degrees.  The rudimentary breast development in men can present as "puffy nipples" and this is easily corrected by resecting the breast tissue behind the nipple and areola.  The cost is reduced from the usual "gynecomastia" correction because the technical requirement is shorter, simpler, and more straightforward. 

How do you tell whether the problem is excessive breast tissue (gynecomastia) vs. excessive nipple size?If you have gynecomastia (enlargement of the male breast), the problem will get better when your nipples are cold (as you described).  Why?  Contraction of the nipple-areola will "push back" on the protruding breast tissues.  If you have excessive nipple size, the problem will actually get worse when your nipples get cold.  Why?  They'll stick out even more.Why is this distinction important?  The treatments are completely different.  For excessive breast tissue, a "real" surgical procedure is required to remove the excessive tissue through a small incision at the junction of the pigmented and non-pigmented skin of the breast.  For excessive nipple size, a "minor" surgical procedure involves "clipping off" the excess tissue.  This is a much simpler procedure with a smaller recovery (and cost).