How can I get rid of puffy nipples without surgery? (photo)

Currently, there is no pill, cream or exercise that can get rid of male breast tissue as safely, and permanently as a surgical chest contouring surgery.
Based on your note and the image you most likely have mild male gynecomastia which can easily be treated in a permanent fashion via male breast reduction which is a safe, short and outpatient procedure.
Man Boobs (aka Gynecomastia) affects up to 65% of men! I guess it’s one of those things we don’t talk about but there’s some really easy options to get this taken care of permanently. 
There are typically two types of surgically correctable gynecomastia. The first, is the result of excess fat accumulation in the chest. This one feels like an even-filled squishy breast. Almost like a female breast. The second, is more excess breast glandular tissue and less of a fatty component. This one feels large but usually you can pinpoint a defined ball or hard/firm area beneath the fat. Oftentimes, it’s a combination of these two and there are some really great options to manage these.
If it’s mainly fat, we have a couple options the first is wide awake flash recovery liposuction which is very well tolerated, doesn’t require going to sleep (but you can if you want to) and the second is a non invasive option like CoolSculpting or CryoLipolysis which is freezing the fat cells.These are both great options for patients which primarily fatty gynecomastia or pseudogynecomastia. 
The other group of patients we typically see, which is the majority, will have some fat but also a rubbery, more firm glandular component which tends to be centralized under the nipple area. For these patients, I like to use the Pull Through Technique along with liposuction. This involves a tiny incision away from the breast, concealed in an area of natural shadow, and through that incision we first liposuction all the fatty tissue and then use that same incision to remove the rubbery glandular tissue from behind the nipple. The scars are imperceptible and heals great. 
Both of these options lead to a masculine and flat chest surface that won't show through sweaters or shirts which is usually the main complaint I hear from patients with gynecomastia

The issue you are describing is most likely due to undesired glandular tissue under the nipple / areola. I do not want to suggest that this is a simple surgery, because I have corrected many gyncomastia surgeries when men have been left with divets, visible scars, etc. However, in the right hands, this is a surgery that should not take much time, and require limited recovery. For some men, this procedure can be absolutely life changing.If a male is in his early teens, or if the puffy nipples have been present for only a few months,, this may be hormone related and may go away.