Transfer from Osaka to Shinjuku/Tokyo

Hi, we will be staying at Swissotel Nankai. As we will just be staying within Osaka then just transfer to Shinjuku middle of our trip, the JR Pass, as per Hyperdia's estimate is not suitable for us. We will just purchase tickets on a per day or per trip basis.

My question concerns our transfer to Shinjuku via shinkansen. I will be traveling with my husband and 3 young kids, and will be coming from Namba Nankai station. For the transfer using JR trains, we will be transferring on April 15, 2014, how do we purchase the shinkansen tickets? Right on the day itself or do we have to purchase in advance? I know these are reserved seat trains, just the ordinary ones. Where do we purchase these? When (the day itself or the day before?) do we make the purchase? We plan to pick a 10am transfer - will that be a suitable time - No longer rush hour? Also, for the purchase of the tickets, will the JR ticket station accept credit card or just cash? It will be our first time in Japan and our first time to use the subway. We will be proceeding to Keio Plaza in Shinjuku.

Thanks very much.

Jocelyn R. Ong