Here's an example of the kind of speedrun you can do.
http://www.youtube.com/watch?v=faKMUSMUpm8
This was on the Anniversary Collection, and it didn't use Navi Mode, but the weapon switching buttons are still there. That means you have more opportunities to use weakness weapons on minor enemies and secondary weaknesses on bosses when other weapons aren't convenient. Remember that 1 through 3 don't have buttons for switching to your transport items, but you'll notice that you can instant-switch to the weapon beside that item to save a few frames on the menu.
By the way, here's what you can get if you could connect to a PocketStation.
http://www.youtube.com/watch?v=mcv10gKQF9A It appears that "leveling up" the Robot Masters gives them larger life bars during the normal game. So you could either level up RockMan and make the game ridiculously easy, or you could level up all the bosses to make their fights ridiculously long. Probably not something you'd want in a speedrun, but it's funny to see once.
