Still no DAC (digital to analog converter) so no true-analog and the same low-quality sound. There are audio boards with an audio chip that includes a DAC and SD card slot, etc., and the Arduino simply acts as a controller. (There are limitations to these things too.... The Arduino can't read the files on the SD card, so it can' read the file names or metadata.)
You'll probably need a multimeter and a mating plug to figure-out the pinout. ...If you buy from a reputable supplier you'd get a link to the datasheet showing the pinout. 
Here are the connections for the mating plug. (With the jack you have, you can't see the connections.)
Or, you can plug-in headphones and randomly connect a battery to figure what connections make a "click".
Since the player is mono you can connect left & right together.
You'll need a resistor in series. The "absolute maximum" current from an Arduino is 40mA. At 5V that works-out 125 Ohms (Ohms Law). That's the minimum load resistance. Headphone impedance varies but 30-60 Ohms is typical so a 120-Ohm (or more) resistor in series will keep it "safe". Two equal resistances/impedances in parallel are cut in half, so with 30-Ohm headphones and left & right connected together that's 15-Ohms.)