Meaning of "type cast solid"

Try parsing it as "a line of type, [which had been] cast solid". It is a line of type for printing, which has been cast "solid": in one piece, instead of the individual letters being inserted in a form and capable of being moved around.

This refers to Linotype, an old hot-metal form of printing. From Wikipedia:

The Linotype typesetting machine ( /ˈlaɪnətaɪp/ lyn-ə-typ) is a "line casting" machine used in printing. The name of the machine comes from the fact that it produces an entire line of metal type at once, hence a line-o'-type, a significant improvement over manual typesetting.

Basically, the machine cast lines of type using molten lead. You could pick them up once they had cooled, stack them together to form paragraphs, etc.