I am 4'11" and wear an H cup bra. I have shoulder, neck and back pain. Should I have a reduction?

You sound like a good candidate for breast reduction based on your bra size and neck , back and shoulder pain.  You do not give your weight, but it is best to shoot for a BMI of close to 30 to minimize surgical complications so weight loss prior to surgery will not fix the breast problem, but will lower your risk of complications.   When you lift your breasts up with your hands and take the weight away, does the neck and back and shoulder pain go away?  If so, breast reduction will likely help you.  I would suggest meeting with the doctor who manages your fibromyalgia before scheduling the surgery.  As you know, surgery may stimulate fibromyalgia symptoms temporarily.  Patients whom I have treated for breast reduction who also have fibromyalgia have still benefitted tremendously from the breast reduction, so I would encourage you!

Thank you for the question. Based on your description you may be an excellent candidate for breast reduction surgery.   However, much more information is needed to determine whether you should proceed with surgery, and if so when (timing).  In my practice, I individualize the “timing” of surgery based on a lot of information that is transmitted during the consultation process.
“Typical” patients who present for breast reduction surgery are women who have disproportionately large breasts, causing problems such as neck/back/shoulder discomfort, postural changes, bra strap grooving, skin irritation/rashes under the breasts, and/or difficulty with activities of daily living and/or exercise etc. There may be both physical as well as psychosocial “stress” caused by the disproportionately large breasts.
Reducing breast tissue mass and elevating the breasts on the chest wall tend to improve or alleviate many of the symptoms associated with the disproportionately large breasts.
Patients considering breast reduction surgery should also consider the potential downsides (risks/complications) associated with the procedure as well. Poor scarring, for example may be associated with the procedure.  Additional surgery may be necessary in the short or longer term for a multitude of reasons.
When the time is right, I would suggest starting with the American Society of Plastic Surgery and/or the Aesthetic Society of Plastic Surgery to obtain a list of well experienced board-certified plastic surgeons.
Then, I would suggest you visit a few surgeons whose practices concentrate on aesthetic surgery. Ask to see lots of examples of their work and preferably speak/see patients who have had similar procedures done.
Then, it will be important that you communicate your goals clearly with your plastic surgeon and make sure that you have realistic expectations prior to proceeding with any type of surgery. In regards to breast size desired, also communicate carefully. In my practice the use of goal pictures are helpful in this regard; a discussion of cup size can be confusing and imprecise.   Therefore, I would not suggest that you communicate your goals and/or base your satisfaction with the outcome of surgery on achieving a specific cup size.  I also find that the use of pictures is more helpful than the words “natural” or "proportionate” etc., which can mean different things to different people.
Many of my patients choose to have enough breast tissue removed to help alleviate symptoms while retaining enough breast tissue to remain proportionate to the remainder of their torso.
I hope this helps.