If this is "regular" Nano with the ATmega chip, the datasheet says output-high should be at least 4.2V with a 5V supply, at 20mA and 85 degrees C. (It drops to 4.1V at 105 degrees).
Output-low is 1V or less.
Connected to an input that's pulled-up to 5V I wouldn't expect any voltage drop (or any current flow).
Since the outputs are driven high or low you don't need pull-ups but they won't hurt. Inputs can float so sometimes they need pull-ups depending on what's driving them.