Help needed in calculating battery life

If I make the LEDs turn on only when a button is pressed, how much will it improve the battery life?

It depends on how many LEDs you have. Realistically though, it's a massive amount. It'd take maybe 50mA to light up 10 normal LEDs at modest brightness, but with them off and the Pico sleeping it'd only be 0.02mA.

I also heard that the moment an LED lights up it sucks a ton of power for a fraction of a second and then stabilizes. is it true?

Honestly I'm not sure, I think it acts a bit like a capacitor. I know that sometimes to get the most efficiency/brightness LEDs are actually driven with Pulses (like those really annoying brake lights on cars that leave a dotty trail on your retina)

Puck

I'm not sure I've seen one but there were a few people looking at adding an OLED display to a Puck.js. There's also a 3D printable case that includes holes for watch straps. It's a bit different, but it could make quite a neat little device.

The same would apply with the display though - you'd need to turn it off most of the time.