We are travelling to Egypt via Dubai mid May. We are flying Cathay Pacific and arrive in terminal 1 at 0600 with an Emirates flight to Cairo leaving from terminal 3 at 0850.
Do you think we have enough time to collect baggage, clear customs etc and make our way to check in with Emirates?
I have read about an meet and greet service called Marhaba which assists you and fast tracks you through the process, at a cost of course. Has anyone used this service or do you think it is unnecessary?

If your flights are on one ticket you will be a transfer passenger and there is no need to go through customs/collect your baggage. Just check in your luggage to your final destination.
But the flights are on separate airlines.
Its possible that your flight into Dubai is a code share, and that they will let you check your baggage through (check with the airline if they will). If not:
Emirates will let you check-in online 24 hours in advance, so you could do that, meaning you only need a baggage drop, which is very quick at Dubai, and can be done up to 1 hour before departure.
At peak times (and 06:00am is peak time) you should assume it will take at least 1 hour to get through passport control, and claim your bags. The two terminals (1 & 3) are right beside each other - its about 700m from door to door, so possible to walk it, although I think they also do shuttle buses.
If your flight is right on time, and everything works, then you just about have time to do it. But, only just. I wouldnt take the risk. Marhaba services are pretty good - they do what it says on the tin, they will fast track you through passport control, which will save you at least half an hour of queuing, and they will arrange ground transportation to the other terminal. Also, the Dubai airport is massive, and coming off a long haul flight, and having to run several kms with bags in order to make your next flight is no fun at all. Marhaba will put you on one of those little golf cart things, and save the hassle of trying to navigate your way around. I think on this occasion they are probably worth the cost, just for peace of mind.