How to power Arduino pro mini and DFPlayer mini togather with 3.7 V LiPo battery?
Use a buck/boost voltage regulator. Those are high efficiency switching regulators, and can either increase or decrease the input voltage. Feed the output of the regulator directly into the Arudino's 5V (or 3.3V) line, bypassing the on-board voltage regulator.
What input voltages do the 2 devices need? Both 5V? If so, you can use a single voltage regulator to power both devices. If one needs 3.3V and the other needs 5V you may have to have 2 different voltage regulators.
By using a switching regulator you can bypass the fairly inefficient linear regulator in your Arduino supply 3.3V or 5V to it directly (depending on the type of board you have.)
Edit:
Ok, I looked it up and a DFPlayer Mini player appears to be a shield designed to plug directly into an Arduino. It looks like it can be powered from 3.2 to 5V, so you should be able to use it with either a 3.3V or 5V Arduino Pro Mini. Once you've decided if you're going to use a 3.3V or 5V Arduino, pick a buck/boost voltage regulator with the desired output voltage (or get an adjustable one and set it appropriately.)