I just lost my waist pouch with all of my IDs, passport, credit cards and I have a UPS delivery coming today.

Will they require ID or do I just need to provide a signature to receive my package?

Whenever DHL delivers at home, they just ask for the signature of whoever collects it, without asking for an ID. If, however, you miss it, and have to pick up the parcel from somewhere else, then they expect either an ID or Driving License.
EDIT: Am assuming UPS is not very different.

I never had to show my ID, just mention out my name or the organisation I represent and sign the papers. But then it is very easy to miss out on receiving a delivery and frequently had to present ID upon pickup.

I see. What type of ID would I need to present if I have to go to the UPS office?

Your residence permit would probably be your best bet, although they'd probably take a passport or driving license as well.

passport or driving license should do the job !!

That's right. The Permit or Passport was acceptable.

Will UPS normally ring your bell and wait for you to come down? Or could they just put the sticker on the door without ringing?

Maybe I should just wait in front of my apartment building for the rest of the day so I don't miss them.

They will tell you eg wait between 10.00am to 12:00 noon, but the guy comes outside these hours more likely to my experience. I got around it by insisting on having the delivery guy's mobile number and since then never missed a delivery.

Me thinks they would prefer an attempted delivery in order to meet their schedules and let the customer pick up.

They usually ring the bell.

Hi.. I missed a delivery from UPS Express and I HAVE to get this by next week no matter what because I am leaving the country... Is there any UPS office in Zürich where I can go in person and pick it up in case I miss it at the second attempt?

I checked the UPS website and there seems to be a store in Rümlang.. but thats too far.. is there any UPS store in Zürich city or its suburbs?

Haven't they provided you with a card/note to say that you've misssed a delivery? Or left a message on your phone?
These will normally give a contact number for you to arrange re-delivery and I'm sure it would also serve to find out if you could collect your parcel from their depot instead.