You can't walk between T1 & T3 these days.

You could (in theory) get an airside bus from T3 to T1 on arrival at DXB - however, they may check your BP so this may not be possible. And once you get to T1, you'd then need to plan your exit route. You can't simply walk out, you'd need to go the wrong way through transit security to join the arrivals stream, and then go through immigration.

You'd have a similar problem at T3. You can go through transit security to get airside, but you'd then need to do the reverse to get back to arrivals.

I think your best bet is just to head straight to T2 and use the lounge there. Or get a hotel for a few hours.