How can I program a STM32 Nucleo board without using the on-board ST-LINK?
Yes you can use an external programmer. First, remove both of the ST-Link jumpers from your Nucleo board to cut the on-board ST-Link from the main microcontroller.
You need these connections for SWD interface:

Power: You need to power the Nucleo board somehow. One can use 5V or 3V3 pins to provide external power. I'm not sure if your ST-Link can provide power. Chinese clones can do that, but the original one may not. The original ST-Link may also require a 3V3 connection to detect the presence of power at target, even though it doesn't use that pin to provide power to the target.
GND
SWDIO: Probably needs to be connected to PA13. Check the datasheet or CubeMX
SWCLK: Probably needs to be connected to PA14. Check the datasheet or CubeMX
nRST: This is optional and depends on tool settings.

But I would prefer obtaining a proper USB A - Mini B cable.