Yes, you can pick up the Nikko World Heritage Pass at Asakusa or Ikebukuro stations.
Will you be wanting to pick up non limited Express trains (journey time approx 140-150 minutes) or do you want to pay the supplement for the Ltd Express trains (approx 110minutes and 1,400yen each way but the pass gives you a 20% discount)
If you want to pay the supplement you can go direct from Asakusa to “Tobu-Nikko” without a transfer.
If not, Asakusa to Hikifune takes 5 minutes (there’s a week day departure at 8:13) There’s then a train that leaves Hikifune at 8:20 am to Minami Kurihashi arriving at 9:20am. From Minami Kurihashi 7 minutes later there’s another train to “Tobu Nikko” arriving at 10:44.
Journey time is 144 minutes (151mins from Asakusa).
This is covered with the Tobu Nikko Heritage pass.
If you’re happy to leave early (and can collect the pass ahead of time) there’s an excellent morning service leaving weekdays at Asakusa 5:58am. It involves one quick change at Minami Kurihashi and arrives at Tobu-Nikko at 8:16am, journey time 138minutes. Or get the 6:44am departure, same transfer, but takes 152 minutes, arrives at 9:16am.
To check schedules go to the hyperdia rail search engine, uncheck the JR Railways box so you are exclusively looking at Private Railways options and uncheck the Limited Express box so you can see the options that don’t involve paying a supplement. Remember to plug in “Tobu-Nikko” as a destination, including the hyphen.

To get from Nikko to Narita airport you can use the Tobu lines to Minami-Kurihashi. Change there for a tobu train to Oshiage and change there for the Keisei line direct to Narita Airport - Oshiage to NRT is with the Keisei rail company, costs an extra 980yen and takes just over an hour, that’s on top of the 2 hour or so Tobu journey from Tobu-Nikko to Oshiage. There are slightly quicker options but they involve more transfers and less convenient train changes.
What I’m not certain about and would enquire after is the boarding at Asakusa and getting off at Oshiage. The Tobu Nikko Pass covers a fixed round trip from Asakusa to the Nikko area, only the train travel in Nikko is unrestricted. I would ask if it is OK. If it isn’t, just get back to Asakusa and get the Keisei Oshiage “Access Narita” line to NRT from the other (separate but partly connected) Asakusa Station. It’s a direct train (labelled as Access-Tokkyu on hyperdia) and takes an hour. Fare is about 1,300yen.