How do you get rid of puffy nipples for men? (photo)

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.  

Based on your pictures your diagnosis is likely gynecomastia. The only reliable (and most effective) management of persistent gynecomastia involves surgical excision. It is often best treated with  partial excision of the prominent glandular tissue  as well as liposuction surgery of the peripheral chest area.  If the prominence of the chest wall is caused by adipose tissue (lipomastia), then liposuction surgery may suffice.
 When the time is right, seek consultation with board-certified plastic surgeons who can demonstrate significant experience helping patients achieve the types of outcomes you will be pleased with.   Best wishes.

Your puffy breasts/nipples are due to condition called gynecomastia, a benign growth of breast tissue. This is usually a result of incomplete resolution of breast growth in boys during puberty. In your case it seems to be a mild deformity which can be easily corrected with surgery in local or twilight anesthesia.In person consultation is recommended. Good luck.