You should be able to obtain the record by contacting the municipal archives in Michalovce or the State Archives branch in Presov. I obtained death certificates for my grandfather who died in Michalovce in Dec 1920 and my great-grandparents who died there a few years later. At the time the charge was $15/record paid to the Slovak archives in WAshington DC.
Presov - not Kosice? Michalovce is in the Kosice region.
My understanding was that only the village archives have the birth records. Michalovce has more than 70 village archives. Since I can't identify the village, this seems to require contacting archives one by one. Many do not respond to emails or pick up the phone. I found my grandmother's birth record this way though.
My grand-father and great-grandparents lived in Nagy-Mihaly, now Michalovce, and the records were in the town hall there. Your message states that the man your are researching was born in Michalovce in 1911. You didn't say that he was born in the Nagymihalyi jaras (district). At the time, the entire area was, of course, part of Hungary. Now it is in the Kosicky district.
Contact information for the Kosice branch of the State Archives is:
Štátny archív v Košiciach
Bacikova ulice c. 1
P. O. Box C-26
04156 Košice
Slovakia
Telephone: 421 (0) 55 6222415
Fax: 421 (0) 55 7293245
E-mail: archiv.ke@minv.sk
Website: http://www.minv.sk/?statny-archiv-v-kosiciach