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.
Bus & Jeep Services are available from Pokhara via Beni, Then Beni to Ghasa, via Tatopani, to Jomsom. It will take rougly 12 to15 hours.
Tatopani is much better to spend the night since you can enjoy the hot springs near the Kali Gandaki river. The accommodation is Tatopani is pretty good (for a trekking destination), but of course not to be compared with the variety of choice and standard as in Pokhara.
Few years ago, I visited Muktinath. Here's my experience in brief. Hope it would help you.
I took a micro bus from Kalanki for pokhara it took about Approx 6 hours then i took a taxi to go to beni(actually the bus was already pack) it took about Approx 5 Hours. Then from beni to tato pani i caught another local bus it was a trip of around 3 Hours to reach there. From tatopani you can catch a local bus to ghasa but jeep is also available as a extra option. The road is about around 2 hours long. From ghasa you must take a local bus or jeep if you are thinking to reach jomsom on your own vehicle then you must leave your vehicle on ghasa unless you are having your own bike. Even the reserve vehicle are not allowed to move forward from there. It will take around 4 hours to reach the destination.
Few years ago i was on a 5 days tour to Muktinath. I didn’t know much about this trip on bus or any other 4 wheeler. i went there on bike. it took 2 days up to Jomsom from ktm. and 2 days down to Ktm. We spend 1 more days to visit Pokhara. At 1st day we reached to Beni and stayed their at night. 2nd day we stayed at Jomsom. 3rd day morning at Muktinath and back to Ghasa at night. On 4th day,we were back to Pokhara at around 4 pm. We visited Pokhara at morning with in 6 hours and back to Ktm.