Using Microphone preamplifier IC Max4468
Does it means that opamp will provide inherent gain of 5, irrespective
  of feedback resistors.(Page 1 of datasheet)


No, it means that you must not choose resistors that produce a gain of 5 or under or the output signal may go unstable.


  Non-inverting terminal should be connected to ground or both
  non-inverting and inverting terminal should be given microphone input.


No, that's not how this circuit works. As shown it appears fine and will have a gain of about -10 (minus means inverting). The non-inverting pin connects to a mid-rail voltage to properly bias the op-amp when using a single supply.