We are older parents as well, but I'm not sure that her not wanting to do things with you is a function of your age. I think it's a function of HER age! I think most teens are preparing (emotionally if not financially or academically) to be on their own in a few years. So becoming independent is a normal evolution. If she has a car (or at least access to yours), she has even more freedom.
What we did was make our home a relatively cool place to hang out. We always had snacks and "real food" available, we had a TV & pingpong set-up in the basement along with a small fridge, and I developed some teen-friendly recipes like sweet & sour meatballs and quesadillas that I could whip up quickly. I would give the kids their space, but I stopped in periodically with food or to pick up empties as a way of giving the place a quick once-over o be sure there was no booze or other illicit substances going on.
The other thing we did was allow our son to bring a friend on vacation - usually we rented a house near the shore, so it wasn't really any extra expense. The kids did their own thing at the beach or we dropped them off for mini-golf, but other times we did things together (golf, bumper cars, going out for dinner or ice cream, etc.).
We also always kept at least one museum membership at the science museum or a local environmental museum, and would go as a family or with one extra kid. We also got passes from the library for museums and other attractions we didn't visit that often. We'd let the kids go off by themselves for part of it but would rendezvous at certain exhibits and to plan our next step.
We also were regular attendees at his events (track meets, awards dinners, etc.) but made sure we weren't the embarrassing parents who yelled out coaching instructions from the stands! We made friends with other parents and, even though we didn't socialize otherwise, we enjoyed seeing each other at subsequent events. That also gave us a good sense of the kids he was hanging out with and what sort of parents were supervising (or not) the groups of kids at parties.
Mostly, although we missed all those fun times together with just us, we learned to accept he natural separation that starts to occur around this age, and used those times to make sure our son "experimented" with his freedom in sensible and safe ways, with us providing necessary (if surreptitious) checks on his wellbeing and safety. Today, he's 28 and very close to us, wanting us to meet his friends and come visit him and so on.
I think the most important thing you can do is not see this as a reflection on you (or your age) but rather her stage of development, and therefore not something to guilt-trip her about. Yes, it's sad, but it's also natural.