Yes you can drive your own jeep or you can take a local bus up to Jomsom.
But I prefer to ride a motorbike up the rugged rocky trail to Jomsom. More than 70% of the road is rocky off road. Riding a motorbike is the best option to reach Jomsom.
Recently, I along with my other friends went for a motorcycle tour up to Jomsom. We were six person in 3 bikes. It took us 2 days to reach up to Jomsom as the road was rocky and we have to be careful while riding.
Since we ride from Kathmandu, our first stop for the night was at Tatopani. There are few hotels, but I urge you to book the hotel first before reaching there, as there will be more tourists in the pick season and there will be less chance of getting hotels. As we were tired from whole day riding our bike, we decided to take a dip in the nature hot water spring at Tatopani, which helped us to relax.
Next morning, we took an early ride to reach Jomsom. It took us about 6-7 hours to reach Jomsom from Tatopani.
The view was awesome throughout our ride. I bet you will not get a chance to see these view while taking the jeep.
So, I recommend you all to take a bike up to the Jomsom for thrilling experience.