will my HD 7870 fit in this micro gtx case?



Hi - Difficult to tell - you will need to measure(or get from Apex site) from the rear panel to the hard drive cage,
you need 225mm/8.9" clearance to fit it in.

You also have another issue; the 300w(21 +12v amps) PSU included with that case will not power a system with an hd7870.