This all depends on what you mean by managing. This can be anywhere between $15- $35 and hour. Or per contract work/job. I suggest browsing some job sites for description on work, weather it will be code or administration and calculate the average.
Manging a Mysql database of a site means to secure, monitor, backup and maintain a mysql database used by a website. This involves all steps on the db end.
So what could be the service charges for such work if website does not deal with huge data, work could be just to update few pages a month and if some security problem occur, solve them as quickly as possible?