Here is pinout for AVR ISP programmer - unsolder battery and power up 3V pins to programmer 5V, MI->MISO programer(Arduino digital Pin 11), MO->MOSI(digital Pin 12), CK->SCK(digital Pin 13) RS->Reset(digital Pin 10) and GD-> programmer GND.

How to flash ATtiny13 with Arduino:

https://www.hackster.io/taunoerik/programming-attiny13-with-arduino-uno-07beba

Ttiny13 core selections:

• Board: ATtiny 13
• BOD: BOD Disabled
• Clock: 1.2 MHz internal osc.
• Compiler LTO: Enable
• Port: (Your Arduino port)
• Programmer: Arduino as ISP

Source code(for Arduino IDE - "12LEDWachATtiny13.ino") here:

https://github.com/techn0man1ac/PerpetualLEDWristwatch/tree/main/LumiTime/Code/12LEDWachATtiny13

This project full open source(PCB and code).