We are arriving into Terminal 2 on the 28th December and want to go to the business lounge in Terminal 1 before getting a flight from Terminal 3 eight hours later.
Dubai Airport currently has three passenger terminals. Terminals 1 and 3 are directly connected with a common transit area, with airside passengers being able to move freely between the terminals without going through immigration, whilst Terminal 2 is built on the opposite end of the airport. For transiting passengers, a shuttle service runs between the terminals, with a journey time of around 20 minutes from Terminal 2 to Terminal 1, and 30 minutes to Terminal 3.
So it appears there is a means to get between the terminals if you're a transiting passenger, but you have to take the bus, not walk.
You should not have any issues traveling between Terminal 1 and 3. As you are arriving at Terminal 2 and have a connecting flight from Terminal 3 you will be put on the transit bus to take you to Terminal 3.
I would like to add a point to @Mark's answer that unless you have a boarding pass for Terminal 3 or 1 (gates A, B or C) you cannot freely move between Terminal 2 and Terminal 1. The airline has to specifically allow you to go there, because you may need to use public transport and they will ask to see your boarding pass before letting you on the bus.
Terminal 1 and 3 are connected, so you can freely walk around; and to get to gates A (the dedicated A380 gates), you need to take the train.
I arrived at Terminal 2 at night and had a big surprise: there is no bus that takes you to T1 or T3, only a taxi. So be prepared with at least 50-65 AED to pay to the taxi driver that will get you to T1 or T3 metro station.
Everyone says the transfer is available only if you have a ticket to fly to another destination. I was very disappointed about this.