So for the runs we have there's a few things of note.
Do this normally, select "You Cowards" when prompted and "I'll be glad to assist." then focus Akami, use Spring Harvest for this, get him to go to one knee then cancel harvest and move (I think you have to move). Then go talk to the boat man and boom, done.
Do the Any% route but then go to the bed roll in the harbor, sleep "until midday tomorrow" for two days. Switch weapon to reverse and go to the Main Plaza. Sleep at the Inn "until midday tomorrow". Head to town to trigger the cutscene and signup trigger. Head back through town (or through the harbor not sure which is faster yet) and join the tournament. Win the tournament.
New Game +
Any%- Skip the intro by going to town, then leave town back to the harbor and talk to the boat man.
Do the Any% route but then follow the New Game credits trigger route. Kill the body guard at the inn since you can't pay. Kill your opponents in the battle tournament since you should have a one hit kill weapon doing New Game +.
I just tried comparing jumping and walking and while jumping may be a small bit faster (about .1 second per 9 seconds) it's not worth doing because if you mess up you lose time because of it.