It has a STC15W204S mcu on the board just above that blue switch.
It came preprogrammed: when I turn it on it shows "12345678" on the display and when a button is pressed it makes an annoying noise from the buzzer and changes the number on it to the number of the button pressed.
The board has RX, TX pins and the 3 other pins w/ connectors for external/onboard control of the TM1638.
How can I program the onboard mcu like an arduino? What software do I need? (I would not expect to be able to use arduino ide)
The STC15W204AS is an 8051 core micro with 4kb of flash and perhaps 128 bytes of ram. You'll need a schematic of the board (or you can ring it out manually) and datasheets for both parts to be able to reprogram the STC part.
BTW, I can tell you the keyboard scanning and led driving is being done with the TM1638. The STC part controls the TM1638 via bit-banged I2C. I have STC code that I've written to control the TM1650, a similar but less capable part if you're interested.
As a final note, there is no way to dump the object code out of an STC part. You have to start with a blank piece of paper design for any rework project such as you're proposing.