Do it, do it, do it!
Make a day of it, go for breakfast/brunch, have a nice coffee, chill out and watch the world go by. Go see your museum/attraction, speak to the staff there, tell them how excited and interested you are to be there, ask for their recommendations on what best to see and experience.
If you're not driving, go for a glass of wine or a cocktail afterwards, have a leisurely browse through any photos you've taken, or whatever... Send a postcard! (Quaint, huh)
Bring a book or wear headphones if you feel self conscious.
I'm 41 and easily do the majority of day trips and holidays abroad by myself - I LOVE it!
It's exciting, it's liberating, I get that its not for everyone but there's such freedom to doing what you want, when you want and not concern yourself with someone else huffing, or getting tired, or wanting to traipse around the shops or whatever.
Worst case scenario, if you hate it, jump in the car or on the train, and come home!
Enjoy yourself, whatever you decide 😊