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I have 2 switches which have redundancy between them, meaning fa0/1 on SW1 is connected to fa0/1 on SW2, and fa0/2 on SW1 is connected to fa0/2 on SW2.

Both of the switches have the same BID, however the MAC address of SW1 is numerically lower, hence making it the root bridge.

Now my question is, on SW2, what determines which of fa0/1 and fa0/2 becomes the RP (Root Port) and the other on blocking state?

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up vote 2 down vote accepted

Assuming no other configuration, it's a deadlock and so SW2 will choose the (local) port with the lowest ID - i.e. fa0/1.

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The least-cost port is chosen. If they are identical, the lower port number is chosen (fa0/1 in your case if they are identical). A good explanation of STP protocol is described here.

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