There is the easy way and the hard way to solve a structural model on a Linux High Performance Compute (HPC) server. The hard way is what you are asking how to do. I almost never do it the hard way, so someone who does might reply and give you better information than I have, but this will at least get you started.
1) a) In Mechanical, click on the Static Structural branch. The Ribbon has an Environment tab. On that ribbon is the button: Write Input File. That is the button to export a file called file.dat that you would move over to the Linux computer.
1) b) You would then issue a batch command to start the solver. The command includes the input filename file.dat.
On a Windows computer, it is something like what is shown below, but of course on the Linux computer it will be different. I put the following line into a file called run.bat in D:\scratch where I put file.dat from part a). In this example, I requested distributed solving over 6 processors.
"C:\Program Files\ANSYS Inc\v211\ansys\bin\winx64\ansys211.exe" -i file.dat -o solve.out -b -dis -np 6
1) c) The results are in file.rst which is very large. There is also a file called file.db that is the database. The text output on the progress of the solution is written to solve.out which you would check first. Move the .rst and .db files back to your local computer. You didn't ask the part d) question, how to read that file into Mechanical. I don't know the answer to part d) but someone who does might reply.
You can practice the above on your local computer and get that working first, then figure out how to do the same on the Linux computer.
The easy way requires that the Linux compute server has Remote Solve Manager installed and your local Windows computer is configured to use Remote Solve Manager. For example, in the image below, the remote compute server is called HPCS04. When I click on the Solve branch of the model, instead of solving on My Computer, I can automatically send the input file to the HPC computer. After the solution is complete, the results are automatically send back to my local computer. It's almost no different than solving locally on My Computer.
I recommend you request that Ansys Remote Solve Manager (RSM) be installed on the Linux computer and you configure your local computer to submit jobs to that.