Hi Andrej and
to the forums!
I'm so glad you were able to rescue this sweet little budgie.
First, I'm sure he's very grateful that you rescued him! However, it's true that he's probably terrified. You should give him two weeks to settle into his new cage and home, since it stresses him to have your hands near him, then just spend time talking to him while you're in the same room as him to make him feel more comfortable. After the two weeks, putting your hand on the outside of his cage to begin to get him acclimated to it is the right step, and then proceeding very slowly from there. It's most likely going to be very slow progress but I'm confident that with time, he'll start to warm up to you!
I would definitely take out the mirror, it's not recommended for the reasons you already said, it can cause him to become obsessed with his reflection and consequently stressed and frustrated that his "friend" isn't there.
You shouldn't let him out for a few weeks, especially not before he's at least comfortable with you standing near his cage. As long as his cage is the minimum size recommended for one budgie (45cm x 45cm x 45cm) then he'll be just fine!
Be sure to read through the forum's many Budgie Articles and "stickies" (threads "stuck" to the top of each subform for easy reference) to ensure you're up to date on all the best practices for budgie care. You'll find information on all your questions (including some you don't know you have yet
) on the forums.
If you have any questions after reading through everything, be sure to ask!