Narita Sky Access Line (JR?) to Asakusa Line (Toei subway ~ Not JR)

My research seems to indicate that I can use the Rail Pass from Narita AIrport to ride the Narita Sky Access Line, which at Oshiage (Sky Tree) "morphs" & continues on as the Asakusa Line (Toei subway). It looks like I can stay onboard and continue to Asakusa (my destination). Is this correct?
Since the Asakusa Line (Toei subway) is NOT part of the JR, can I simply use a SUICA card to pay when I exit Asakusa subway station, or do I need to actually buy a ticket at Narita Airport?
My plane arrives in Narita in the late afternoon, so I anticipate traveling with my suitcases during rush hour, so I'm trying to keep things as simple as possible.
~ Ninjakat