OK. Post a link to your player so it is clear what the minimum voltage is.
Also, it appears that your circuit is also dependent on the Arduino Uno being powered (by something) to give you 3.3 volts to drive the switching transistor. So it is not completely clear what role the 3.7volt battery has. Or are you intending at some stage to make the MP3 player into a stand-alone device ?
If power saving is an issue, then you should also focus on the Arduino Uno which, also when doing nothing, happily takes c. 50 mA.