Mechanical engineer studies subjects like Machine Design,Mechanics, IC Engines fluid mechanics which are a essential parts of Aerospace industry as Aerospace Engineering is nothing but the application of same basic principles of structural mechanics, thermodynamics and fluid mechanics in aircraft and spacecraft.Apart from this, you can take-up some internships or projects during your mechanical engineering that include application into aerospace field, which will give you some idea of what more you need to learn.Learn ANSYS (structural, thermal and fluid simulations), any 3D modeling tool, and MATLAB, and you are eligible for any aerospace job.

Hope this helps!!!

Yes, a mechanical engineer student can go for aerospace industry if the student has required set of skills. ISRO recruits mechanical engineers through its ICRB exam to work in the field of aerospace and their are many other companies as well. If you are interested you can also go for any course related to it.

Good Luck!!