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eShop is a free service. The online you would have to pay for would be if you wanted to battle your friend over the internet rather than in person. It would also be used for other games like Splatoon 2 or Super Smash Bros., but some games don't require it like Fortnite and Rocket League.
If you were to buy a physical copy of Sword, you could then purchase the DLC through the eShop without paying for Nintendo Online, yes. You could also look out for a physical copy that already has the DLC preloaded; this is indicated by a different box, as seen here.
