Assuming you have your on ward ticket, nothing stopping you going out and getting onto your next flight, if you want to be extra safe, you can always grab a Expo ticket during your stop on EK flight and near the entrance is a Expo Covid test place which gives results in 4 or 6 hours. Do your test and then go out into Dubai, either go to the Expo, enjoy a meal.
If your planning on staying in, what I would recomend is go to Lounge B in Terminal 3 concourse B. Just say you have a long stay and you will scan your card when the 3 hours expires. Prebook a shower and they will take you to the Int Hotel for a shower session when you are ready to go. As for the sleep pod's, you can do the same, just say you want to scan for two visits. They wont allow you to prebook but have ample pods availible so wont be a issue.