Ok, first of all, are you running the circuit off batteries or a wall wart?
Second, will you need to power the mp3 player or will it run off its own battery?
There are several elements of the project. They are:
1. Wiring a button to the MP3 player (doorbell button in picture)
2. Amplifying the MP3 player's output and sending to a speaker (speaker and amplifier circuit)
3. Powering the amp/speaker (Wall wart or battery, and possibly a voltage regulator)
4. Powering the MP3 player (3.5V regulator in picture, or use the MP3 player's own battery)
Step 1 is technically simple but will require a bit of delicate soldering. You'd have to locate the play/pause button, or the power on/off button on the MP3 player's circuit board and solder a set of wires across it to connect to the external button.
Step 2, you could use a LM386 audio amp IC, a couple capacitors and resistors and a small speaker. The picture shows what looks like an amplifier PCB from some radio or something. Or you could get one of those cheap MP3 player speakers, or even use a set of amplified PC speakers. If you build an amp, provide a 1/8" stereo plug to plug into the MP3 player.
Step 3, powering the amp, depends on the amp you're using. A LM386 can be run off a 9V battery or wall wart adapter.
Step 4, powering the MP3 player. The easiest way is to use the MP3 player's battery. If you want to power it from the same source as the amplifier, you'll most likely need to drop the voltage to whatever the MP3 player uses (probably 3-5 volts), using a voltage regulator. You could use a 78L05 if you need 5 volts, or a LM317 circuit adjusted to whatever voltage you need.