overhead in accessing non-aligned memory

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Lets say the processor accesses 2 bytes at a time, while accessing non-aligned memory which starts at position 1 rather than 0 in the underlying memory representation:

     +----------------+ 
   0 |                | 
     +----------------+
   1 |                |
     +----------------+
   2 |                |
     +----------------+
   3 |                |
     +----------------+
   4 |                |
     +----------------+

What additional overhead the processor will have to bear (which wouldn't have to be done had this access been started from memory position 0 instead of 1)? Plus Is this overhead same if processor could access 4 bytes at a time?

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