MS Excel: How to use format painter with just the keyboard?

Press F5 or Ctrl+G (Go To), and type in the address of your destination cell. You can F5 around and when you are done press ESC to get out of format painter.

Update (November 2020): This Method does not work in the current version of Excel, as it will copy both format and content of the origin cell. Unless you are formatting empty cells, I recommend using the method of WireGuy instead: Use F5 or Ctrl+G to navigate to the target cell.

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Original Answer: It is quite cumbersome, but after you undo (Ctrl + z) formatting the neighbouring cell, you can move around the cursor with the keyboard arrows and press enter finally to paint the correct cell.

If you hold down shift while navigating (before pressing enter) you can paint multiple (adjacent) cells at once.