If you don't send luggage by takkyubin, you're going to need to come back to it at the end of your day before going to the airport. So, here's an idea.

Go to JR Osaka and store luggage there. (It would be nice if someone would chime in about that possibility. I'm sure that there are lockers there, but don't know where or how big they are). From Osaka station, walk/taxi to Umeda Sky building and eat lunch there. Back to Osaka/Umeda area and take Midosuji subway to Shinsaibashi, then walk the shopping street from Shinsaibashi to Nanba. Nanba is a great Osaka experience, different than Shinjuku. Eat dinner in Nanba, then take subway back to Osaka station (about 15 minute ride) to pick up luggage and go to airport by train or bus.