How do I program an LCD screen and do I need an Arduino
OLED screen is really good looking for both indoor and outdoor and I think you have made a good choice.
You need the following things to complete your project easily.

Display (which you have already found one
Arduino (why? easy to program, lots of libraries available, and also tutorials)
RTC - Real-time clock - This is another IC (DS1307, for example)

Arduino need not be UNO. You can use ATTiny85 MCU. the programming will be the same as how you program a UNO. ATtiny85 has no builtin I2C, but the libraries are there to emulate I2C to be connected to OLED displays.
If you want to avoid the third option, go with internal timekeeping alone. But, over a week, you will see the time difference (the internal clocks of the MCU are not accurate, so the time will lead or lag over a period of time)
have you thought about a keypad or a single button to give the option to the user to set the time?