In Premiere Pro you have two different options to fade in music (or fade it out):
Fade Music with Crossfade Audio Transition
The easiest way is to select the audio clip and then choose in the Premiere Pro Menu Sequence->Apply Audio Transition - or use the keyboard shortcut Shift+Command+D (macOS) or Shift+Ctrl+D (Windows). This will add the Constant Power Cross Fade effect to the start and then end of the clip. You can easily change its duration by clicking and dragging the end of the transition and remove it with the Del key. Cross Fade transitions can also be used to blend between two audio files. But if there is no other audio clip directly after your music, it will do a normal fade out instead.
Fade Music with Clip Keyframes
Instead of using a Cross Fade Effect, you can also keyframe the volume of the music clip to create a fade. Premiere Pro has two kinds of volume keyframes: Track Keyframes and Clip Keyframes. The difference is that Clip Keyframes move with your music clip if you move the clip in the sequence, whereas Track Keyframes always stay where they are. Therefore, make sure that Clip Keyframes are enabled and then simply use the Pen Tool to add volume keyframes.
Change Music Duration
Instead of just fading the music out where you want it to stop, you can also edit the music such that it has the desired length. The idea is that songs usually consist of sections that you can duplicate, delete or rearrange to make the song longer or shorter. For this approach is is crucial to place your cuts exactly at the beats of the music. This can be does easily with our Premiere Pro extension BeatEdit.
The workflow is shown in full detail in this video tutorial.