I bank with Credit Suisse and have a standard Maestro card. Because my residence permit has not arrived yet, I don't qualify for a credit card.

My question is this: Is there any way to use my Maestro card to make payments online to buy cinema tickets and other stuff by mail order?

I use online banking, so I am quite familiar with the standard online payment methods, but for some things (like cinema tickets) this would not work.

Any suggestions / comments gratefully received!

Don't think its possible. At least I've never figured it out.
Just got my permit though, so can finally get a credit card...woot.

No it is not possible.
Maestro is an example of EFTPOS. In order to perform this function you need to have a key to the account. The key is two factor your card and your pin number. Without these there is no way they can make the authorization... This is very subtly different to credit cards...

You might want to check out the post ofice card / www.postfinance.ch
Within Switzerland, a lot of online shopping can be done, tickets etc. can be purchased directly with the "direct debit" function.
I don't know if you need residency permit for it though...

This summer I asked whether I, a tourist (i.e. non-resident), could open a postfinance account. I was told yes, but I would not have a postcard.

This was at the post office in a small village, he may have been mistaken. I did not pursue the issue, so am not totally sure.