In your situation, I'd probably go into Narita City immediately after arriving at Terminal 1. It seems to me that would be less hassle than shuffling between two terminals and going in and out of secure zones at the airport. Besides, seeing the city immediately relieves one of any stress or concern about getting back to the airport in good time.

I'd take one of the ordinary Keisei trains to Keisei Narita station and leave my hand luggage in a coin locker and take a walk around town. Then I might take a cab to the Yamatonoyu onsen for a bath and lunch and then pick up my bag at the station and take a train to Narita Terminal 2.

At the airport, I doub't I'd bother with a day room. The shower facilities etc. are OK (but why bother, if you've been to an onsen?). The day rooms themselves are rather spartan cells with the muffled sounds of a busy airport as background noise, that don't feel very relaxing to me. If I wanted a nap, I'd probably head for Rafine and sign up for 60 minutes massage. Sure, it's more expensive and it is in an open area rather than a closed room, but I generally fall fast asleep and wake up feeling great.

I've used the shower rooms at NRT once in the past but usually I have access to the ANA lounge which offers showers and reclining chairs in a curtained off environment that makes it possible to have a nap.