Hikes you can do without renting a car
Unfortunately in many cases you need your own transport to reach the hiking spots, and renting a car is cheaper than going on an organised tour. However, if you can't or don't want to rent a car, there are still several hikes you can do without it. Following is the list of these hikes by city.
Riyadh
All the hiking locations are quite far away from the city: the nearest is the Camel Trail, but it is still around 50 kilometres. A ride-hailing service, such as Kaiian or Careem, would cost around SAR 80-100 one way to Qusur Al Muqbil, from where you can get to the trailhead on foot. Getting back would be more difficult, though. You may or may not get lucky to find another car to take you back. You could try hitchhiking, but then again, you need luck. So it is not the most convenient or cheapest to do this hike without your own transport, but perhaps it is possible.
Abha
One of the best hikes in Saudi Arabia is the Al-Qarun Trail in the Soudah mountains, and this is a hike that you can do without your own transport. In fact, even if you have a car, it is not too convenient, as you have to make your way back to the spot you parked your vehicle. The trailhead is about 25 kilometres from the city centre, and you can get a ride-hailing service for around SAR 30-40 to this spot. However, once you descended all the way, getting back is a bit more difficult. You may have to walk to the main road, which is an extra 5 kilometres, but even there you may not find a ride-hailing car. Your best bet is to hitchhike. You need patience, but it will work out.
Taif
The Al-Hada mountains are only about 25 kilometres from the city centre, so you could possibly do the Mecca view hike with a ride-hailing service. The problem here is getting back, as the chance of finding another car in that area is very slim. You have to be prepared to hitchhike if you want to do this hike without your own transport.
Medina
Both hikes mentioned, Mount Sela and Mount Uhud, are within the city, so here you won't have any issues doing them without your own transport.
Al-Ula
All three hikes mentioned, the Wadi Al-Naam Trail, the Mahash Fort Trail, and the Dadan Trail, are relatively near the city centre (about 5 to 15 kilometres) and can be reached by a ride-hailing service. You can get another car back for sure, but you may have to wait a bit as they are not always available. You can also try hitchhiking, and even walking back is not impossible.