Looks like these are
ways to measure time, listed roughly in chronological order of their invention.
The first and oldest, strong and bright / Performs his duty dawn to night.
Looking at the sun will give the approximate time of day
Inside a curve a humble stick /
Does even better - what a trick!
The sundial gives a more accurate time than just looking at the sun. The simplest sundial is just a curve, where the locations of the shadow of a fixed stick (the gnomon) are marked.
Many turnings, night and day, /
Dust that runs a narrow way.
An hourglass has sand running in it, but you need to keep flipping it over when it runs out.
Escapement, weight and turning part /
Co-operate in ancient art.
A mechanical clock has an escapement, a weight (the pendulum), and clock hands that turn
Resonant geometry /
A quartz clock works by running electricity to a quartz crystal, which responds piezoelectrically, and starts to vibrate in a very predictable manner, which gives a very accurate way to measure time.
Small and mighty, they're the best /
When chilled. They govern all the rest.
Atomic clocks are currently the best way to measure time, the time they keep is considered definitive, and all other clocks are synchronised to them. Chilling an atomic clock (eg. with liquid nitrogen, or even close to the absolute zero with lasers) reduces the background noise and slows the atoms down. Both of the effects improve the accuracy of measurement.
That would make the title into
old-timer and egg timer (This is probably what the crypto speaking @jafe is suggesting below)
And OP told me the answer
in a timely manner, one clue at a time, and probably in his spare time.
Kudos to Level 51 for spotting the acrostic.