Thank you for your answer and sorry for my late update.
I was travelling in August as described and it was very easy.
I arrived at Terminal 3 with the Emirates flight. After passing security I was going with the line to Terminal 3 transfers.
It was no problem as there is no more boarding pass check or anything else. Just the normal arrival / transfer security check of you and your hand luggage.
After arriving at T3 I went to the Sleep Pods (which are included in Priority Pass). It was around 7 or 8am in the morning and there were about 1/3 of the pods empty. They will give you 2hrs free but it's no problem to extend the time (hint: they know how, just ask). But it's only the pod, if you want some sheets or something else, you have to pay them (I think it was around $13 or more). And its very cold in there, so take something to cover up with you. The pods are ok for some hours of sleep but nothing special. There is no toilett or shower inside, you have to leave and go to the normal airport facilities.
After sleeping we went to the Alan@B lounge which is on top of the McDonalds. Also no problem there. The lounge is small but nice and has some food and drink options. It was around 12am and there were only 4-5 other people. As PP member they give you a special flyer with the included drinks. They have nice chairs and also massage chairs. I can recommend the lounge for even longer layovers.
About 1 hour before boarding we went down to the transfer bus to T2. As described you have to go through security check the other way. But there are signs there indicating you are allowed to do and also the staff knows. The security line was completly empty and so it was also no problem to walk through.
The bus transfer takes about 20minutes I would say. Arriving at T2 you have to go through security before you can enter the departure hall. It was about 20min before departure and we went to the Marhaba (PP) lounge there. Small food selection, the drink selection is ok but the lounge is completly crowded. There were no free seats and we had luck that someone left. Not really a lounge where you want to stay for a longer time. Good just the get some food and drinks before boarding.
Btw. the flydubai flight to JRO was planed with the 737 MAX 8 but changed to an old 737 for known reasons. It was a really bad flight. They told us there will be an entertainment system, so I was not saving movies to my laptop. But the entertainment system was over 20 years old and the touch screen didn't work on 90% of the seats. So impossible to watch something and I ended up in an 7hrs flight trying to sleep. On the flight back they informed us that the system is out of order today, but I was well prepared
Additional information:
At JRO airport we were not able to use the lounge because of the domestic flight to Zanzibar. They have only a lounge for international flights and you are not able to go there.
At ZNZ airport we used the PP lounge (which is also used by Qatar, KLM and some other, because it's the only lounge there). It was pretty small and selection of food was poor. They had some softdrinks, beere and coffee/tea. Was ok while waiting for boarding but not for longer time and also not if you need to pay.