I would prefer
Soll ich Dr. Mueller schreiben, um an meinen Brief zu erinnern?
It is also possible to use
Soll ich an Dr. Mueller schreiben, um an meinen Brief zu erinnern?
You can use jemandem schreiben (Dativ) or an jemanden schreiben (an + Akkusativ).
It must be "…, um an meinen Brief zu erinnern".
Especially in this example sentence the first variant is stylistically better because it avoids repeating the word "an". (suggested in phipsgabler's comment)
Note: The correct spelling of the name might be "Mueller" or "Müller". I adhere to the spelling from the question because it may be the correct spelling of the person's name, which is not known here. The spelling "Müller" is more common, but it can be written as "Mueller" if the letter "ü" is not available.