Should I design 2 different antennas for uplink and downlink for CubeSat?

I am last year undergraduate student and for my thesis I have to design antennas for cubesat. Should I design 2 different antennas for uplink and downlink or should I use duplexer to use same antenna. In that case how I will design microstip patch antenna for 2 frequencies using duplexer.

There is not enogh information for yes or no answer to this. You have to ask youself some questions before asking for help
What are the uplink and downlink frequencies. If the frequencies are close spaced a duplexer may need to have a lot of mass to give the required isolation, and one thing that is really bad in spacecraft is mass.
How much isolation is needed beteen the TX and RX is needed for the RX no to be overloaded; this will depend on TX power
Does the antenna have to be a patch, a potential problem if most or all of the outside is covered in solar cells or sensors. What happens if the craft is spinning (it will be) and the antenna is pointing at the ground station for only a short period each rotation, so another question you will ne to consider, what is the spin axis in reation to the earth.
It is a non ideal situation you are going to have to be inventive to overcome these real world problems.

Actually the uplink frequency will 5.1 GHz and downlink will be 3.7 GHz. I have started designing a microstrip dual band patch antenna in CST. But I can't set these 2 frequencies. Can you help how to design a dual band patch antenna in CST.

Sorry can't help much there I'm not an antenna designer and I've never used CST.

Instead of a dual-band antenna I would make two separate patch antennas, which gives better isolation between bands, higher gain, and less headache designing them.
At those frequencies their dimensions should be fine for your application. Have to put the antennas on the same plane.