I think that IS your problem....that it IS snapping to the grid.....you want it to snap to the middle of a pixel.....and I've bad news....It ain't gonna happen.
With 'Snaps' on and set to 'grid' you'll have to click in the top left corner of a pixel to get it to fill correctly....and even then its no guarantee. Clicking the 'middle' of a pixel is just a random lottery as to which pixel gets filled.....if at all.
You should find though that at pixel level you don't really need 'snaps' on anyway.....if you have the grid visible its easy enough to place down pixels accurately without needing any 'snapping'.
I'm a bit unsure as to why you want to do this pixel by pixel anyway....even at its original size it would take some considerable time to do but I think you want to make it ever larger? (Its not exactly clear what you want to do to be honest).
If you just want a larger version then re-sample it.
If you want to re-create it then by what method?
If you want to increase the resolution then just type it in...
What exactly is it you want to do?
Don't take that the wrong way.....its just a bit confusing from your first post, it can be interpreted in many different ways especially as the terms you've used do not really correlate with the question.
If you could clarify that would be good. 
(I'll start you off by saying that 'resolution' does not mean how big an image is! 
)
Regards.
MrToM.
