Is there a way to use the atmega 32u4 breakout board to make a mp3 player without the mp3 shield?

Is there a way to use the atmega 32u4 breakout board to make a mp3 player without the mp3 shield?

You don't.
A basic ATmega lacks the computational (and perhaps memory also) resources for mp3 decoding. And even if you do decode to a linear PCM format, you do not have a good way to output it.
You can playback linear PCM data by pulse width modulating a pin, but the limited clock rate will result in limited resolution, and thus poor fidelity. You will also end up with a low sample rate, making it hard to avoid audible artifacts.
The situation can actually be somewhat better with an ATtiny85 - still no MP3 decoding of course, but you can clock the counter/time block at a higher rate and so playback PWM audio with higher fidelity, provided you are willing to tolerate the inaccuracy of the onboard clock oscillator that may shift your audio frequencies by a small amount.
If you want to do MP3, you will need either a more powerful processor (often no more expensive than the one you have, and typically cheaper as a module than anything "-duino"), or else a dedicated function chip to act as a coprocessor for that purpose.