I installed JA2 (which started fine) and then installed JA2 1.13 (7435) and Afterwards the Update (7609).

Now when I start I can hear the sound of video and menu, but cannot see it.
When Alt-Tabbing a bit, or using the Task Manager, I can get it to see the menu, but The mouse is not working too.

Any ideas how I can get the old gem to run right?

What worked for me is i deleted the intro.slf file (i think that's the name) then set "play intro" to false in the INI editor

It should be sufficient to either delete the file intro.slf respectively move it out of reach(to different folder)
or disable the videos in the ini editor.

However you may experience lags with 7609 on win 7,8&10,
in case check thread "Game runs slow under windows 10" on this board for some options to fix that.

You can use Window_8_Fix.zip from https://storage.rcs-rds.ro/links/4729f8d6-f44b-42b7-aa3e-e0ddc6deead6?path=%2FJA_2%2FPatches%2FOther (just unpack into game folder).

It solves both issues with intros black screen and in-game lags.

Tested on original JA2 from GOG.

solution from r0flc0pter post solved the problem for me.
I just renamed the file and that was enough