Can I schedule startup and shutdown with different settings for weekdays and weekends?

Everyday, I usually start my Mac Mini at 16:00 and shutdown it at 2:00 using the Energy Saver schedule settings.

I would like to be able to set a different behavior on the Weekends starting Mac Mini at 9:00 for example. Do you know any way to achieve that (third party software, cron job, superuser trick)?

You don't need 3rd party software. The OS X utility pmset allows you to schedule sleep/wake taking day of week into effect. I haven't tested it, but it appears you cannot schedule multiple repeating events on different times. You'll need to use cron to complete this task.

from man pmset

Schedules the system to automatically wake from sleep on 
July 4, 2009, at 8PM.

pmset schedule wake "07/04/09 20:00:00"

Schedules a repeating shutdown to occur each day, Tuesday through 
Saturday, at 11AM.

pmset repeat shutdown TWRFS 11:00:00

Consider using DssW Power Manager. With Power Manager, you can create multiple separate start up and shut down times for weekdays and weekends.

Power Manager goes far beyond the capabilities of pmset. How to schedule your Mac to shut down (screen recording on YouTube).

Disclaimer: I work for DssW.