I have never seen an Alemite in person, with the original glass intact, to know what the original factory padding was. The assumption is that there would have been a felt-like, possibly braided, material in between the glass and metal. These most likely deteriorated or were swapped out over the years.
The metal window flange on the inside top of the cabinet is slightly deeper than the bottoms, so that you can slide the glass up and in and down. Please, let us know if you find a good rubber substitute.
We would love to see pictures of yours restored, when you are done!