Electrical – Replace switch with timer, instructions don’t match existing wiring

The dimmer you are replacing is a 3-way dimmer, but the timer is a single-pole. They both have 4 wires because the dimmer doesn't need a neutral, but the timer does (to power the timer). You'll need to get a 3-way timer.

Do I need to use the short neutral that came with the new switch

Yes.

(if so, specific instructions are appreciated)

The best instructions you should follow are those of the manufacturor of that switch.

But generally here's what you do.

1. Remove the orange wire cap and add one end of the short neutral that came with the switch you bought. (It's called a pigtail btw)
2. Connect the other end of that pigtail to the neutral terminal of the smart switch.
3. Take the two black wires out of the old switch, and use a volt meter to determine which one is live when the breaker is on.
4. Label it line, or 'from panel' if you want to be extra nice.
5. Connect tha live black wire to the 'line' side of the smart switch.
6. Connect the dead black one to the 'load' side of the smart switch.

This diagram shows the general idea. Ignore the shape of the hole in the wall. Also your switch doesn't have wires coming out of it it sounds, or you'd probably have a red one for the switched load.