Looking to build a stopwatch/timer

So I have this wristwatch. It can act as a 6-number stopwatch and a timer. I'm thinking....replicating this can't be too bad, right? I just don't know where to start/what I need.

I have transistors, resistors etc, but no microcontroller. What do i actually need to make the stopwatch/timer?

A microcontroller would be easiest. Logic ICs would be more difficult. Discreet transistors and resistors should only be attempted by an experienced fanatic electronics cultist!
All the methods above will result in an item which is much larger and expensive than a regular wristwatch. I assume you are going to do this for the fun of it and not for any practical reason.

I don't understand how it works though. Each of the numbers has 7 positions, and the microcontroller only has a few output pins. What if you want to control 6 numbers? How is it actually able to control what numbers are displayed?

By Multiplexing.
Google that.