What should an unofficial transcript consist of? My university only issues an official transcript on paper ( Will take me a month to get it). However, we have a separate student's portal where my "grade history" can be accessed. It is a list of all courses along with the grades and my CGPA, but does not contain anything more than that. Would this work while sending it to a  professor that needs to see it for grad school?
Yes, that's probably fine.  The list of courses, grades and GPA is about all that an official transcript contains anyway.

You can include a note to the professor asking them to contact you if they need anything else.