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 male breast is composed of glandular tissue and fatty tissue. An excess in either type of tissue can cause the chest to take on the look of gynecomastia. Gynecomastia can affect #men of all different body types. There are different types of #gynecomastia. Several common factors can cause gynecomastia including excessive levels of #estrogen, natural #hormone changes, use of recreational drugs or alcohol, medications and their associated side effects, and, various health conditions. Also, there are different types of #gynecomastia."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.

Puffy nipples can be reduced by removing extra breast tissue directly below the areola and nipple. It is often done under local anesthesia as an office procedure. Addtional breast tissue fullness outside of the nipple area is usally removed with liposuction. Exercise will not help this condition.