I have this circuit(attachment). Is it possible somehow to get a 1:2 duty cycle?

 

Welcome to AAC!
You can use a third resistor, in series with and of similar value to R, and bypass one or other with a diode.

 

Usually an odd number of inverters are used.
See here:
http://www.fairchildsemi.com/an/AN/AN-118.pdf

And yes, you can get 1:2 duty cycle with a diode.

Edit: New owner took down that link. I have attached a pdf.