How do I mount rear derailleur on frame without separate hook part?
Some Tourney derailleurs have a sort of extension ('hook' as you call it) that your new Acera derailleur does not. That is fine. The Acera derailleur simply bolts to the derailleur hanger on the frame (the piece if metal pointing downward in your first picture)

Your new Acera derailleur has what is known as a 'B pivot' which I believe the Tourney derailleur lacks. The B pivot is located where the derailleur bolts to the frame. The entire derailleur rotates around the B pivot, the B pivot spring tries to rotate the derailleur anti-clockwise underneath the cassette.

(There is also a 'A pivot', around which the jockey wheel cage. rotates.)

The blue painted screw you mention is the 'B screw', which is used to adjust the position of the derailleur about the B pivot. It is likely that your B screw simply needs to be adjusted properly so that the derailleur is moved away from the cassette

You can look up instructions on how correctly the adjust the B screw. Park tool has a good video on derailleur adjustment which includes B screw adjustment.