I inherited an older Asus VE278 monitor. Is it safe to connect it to a new MacBook Pro with MI? If so, what do I need?

That 1920 by 1080 display does not have an HDMI input, and

that cable only connects from ThunderBolt-3/USB-C computer-end to HDMI display end. It will not function in reverse.

You want a different function, which would be better provided by USB-C to DisplayPort cable such as this one:

Monoprice USB Type-C to DisplayPort 3.1 Cable - 5Gbps, Active, 4K@60Hz, Black, 6ft --Product # 24444 -- $22.49

for slightly more money you could have an adapter and use a DisplayPort to DisplayPort cable.