Solo female travelor Sardinia

Well, I've never been to Olbia but it looks like a nice interesting town. If you plan to stay in town I can't see why you'd need a car.
Except that there don't seem to be many easily reachable beach.
Maybe find a nice hotel with a good pool area?

I would not worry too much about safety, just the general common sense precautions. The only crime warnings I saw were the usual things like not leaving stuff in your car. At one rental, we were told to leave the key under the mat, so that tells you a little bit I think.
Most important is the transport issue. You don't have to have a car, but it does help tremendously and will determine the itinerary. Decide if you are up for it first, and if not, delve into bus schedules, stat! You could go to Cala Gonone and take advantage of boats to get to beaches, and book some day tours for some history exploration (to Galtelli and Dorgali for example). There is a lot concentrated in this area--caves, nuraghi sites, "giant's tomb".
I also flew into Olbia, but I did not concentrate on the north. You can get beaches up there for sure, but I don't know how easy public transport would be (the driving in the north is not intense like in the central/eastern part, so maybe something to consider). For some aspects of history and culture, you could look at the Azrachena and Tempo Pausania areas.

Olbia is a pleasant place to stay but not that easy to get to the best beaches - one issue in Sardegna is that the towns are often not right on the beach.
For just a few days if you don't want to rent a car there's a bus service to San Teodoro. Once in San Teodoro there's a free bus service from the various hamlets to the main beach La Cinta and the Beach Bus to other famous beaches like Cala Brandinchi - some beaches require booking.
More info here:https://santeodoroturismo.it/info-utili/come-muoversi/
If you want to hire a car and concentrate on northern Sardegna there's a host of options. Here's our trip last May to give you some ideas, and you could add Tempio Pausania as mentioned above.
https://slowtravelitalyspain.blogspot.com/2024/07/sardinia-in-may-trip-report.html
Sardegna is a very safe place, no issues for travelling alone. Temperatures might already be high for hiking though, and make sure you have water with you and know where you are going. Inland Sardinia is not a place for solo hiking if you don't know the area. There are nice easy hikes along the coast south of Santa Teresa Gallura

When we went to Sardinia we looked into public transport and decided hiring a car was much easier. There's not a whole lot in Olbia- but we enjoyed some nice beaches up in Santa Teresa di Gallura and some nearby hiking out to Capo Testa. We also really enjoyed the beach in Alghero, which is on the opposite side of the island from Olbia but not a very long drive. The old town is amazing and was founded when that part of Sardinia was part of the Catalan empire making it quite unique within Italy. I felt very safe the whole time I was there, I would just take the usual precautions when travelling. Good luck with your trip.

Thanks so much for the helpful info!