The good thing about domestic travel in Japan is that there is no need to be there early enough to worry about a lounge. Security lines are only ever long during Japanese holiday periods, otherwise you'll get through in 5 mins most of the time, there's even a special line for people with imminent departures but I wouldn't recommend pushing it that far because the HND domestic terminals are quite big and take a good bit of time if your flight is all the way at one end.

As in most countries, Japan international departures are after clearing exit immigration, there's no way to access that terminal's lounges with out an international ticket. There used to be a T3 landside contract lounge on Priority Pass that was fabulously craptastic, it would not have warranted going early for in any situation, especially if then having to move over to the domestic terminal (which is not close). Also from what I've heard, even if you could access the ANA lounge, they frequently reject Priority Pass holders due to overcrowding. Japan has an system on both ANA and JAL where if you make mid-tier once (Platinum on NH, Sapphire on JL) then you can sign up for a credit card that has about a $100 a year annual fee and keep your *G / OWS for as long as you keep paying the card's annual fee. So the business lounges are ridiculously crowded quite often. I've even seen JL's domestic First lounge packed to the gills once in FUK, not sure how that happens but there you go, my companion and I actually went to the Sakura lounge instead as it was mildly less crowded and we could at least get 2 seats next to each other... the staff apologized profusely for the inconvenience of having to use the business lounge, was mildly humorous but no idea why they have so many OWEs to start with to pack a First lounge.

JL's Class J does credit as business class on partner frequent flyer programs, but otherwise it's just a better seat. Service is the same as economy, and no perks like lounge, priority check-in, etc. If you're going to Nagoya city, why not just take the train? I actually do fly to Osaka frequently myself, but it's a ridiculous venture honestly just to get a few extra miles, the train is more convenient by nearly any metric and by the time you factor in transportation to/from each airport the timing is nearly the same. If you were connecting from somewhere else, it would make sense, but going city to city for Nagoya, Kyoto, or Osaka, the train wins out.