You are saying the exact correct steps of how to copy the active window, opening the app, and then to paste that captured image.
But you say you see "Paste" grayed out.
Then you mention, in a later, message that CTRL-V does work.
?? CTRL-V and the Edit>Paste command are the same thing.
I suspect that your first step, the ALT-Print Scrn, is not what you had really done. You may think you did it, but somehow you managed to clear out the image buffer just before your Paste function (if the image buffer was properly captured, in the first place). Or else -- somehow you wiped out the image in the buffer.
For example, if you had some image captured, via Alt+Print Scrn, and then somehow had accidentally entered a Ctrl-C (some text selected and copied) that would wipe out the image buffer, and Paint (or any other image tool) will grey out the Paste function.
Your buffer can only hold onto one copied item at a time, either copied text or an image. The most recent copy (for example a Ctrl-C or some Print Scrn copying function) wipes out the previous item copied.
(By the way, do not confuse the special tools, like the Clipboard, or other clipboard managers, with the buffer. They act like you have multiple buffers to copy and paste from, but in reality, they are just using software to increase the functionality of that ONE buffer, that Windows provides.)
Any of those screen capture tools mentioned by the other respondents are fine tools, and they may simplify the steps that you need to perform your window captured that you wanted to do. But they don't address your originally stated problem of why was "Paste" grayed out?
I hope I have done so for you. Go back and check your steps, try it again. If the Ctrl-V worked, then Edit>Paste should also work.