Can I combine an alarplasty with Italian lip lift? And how many days would it take to look publicly acceptable?

Yes, they can both be done together.  It is a good combination as the same incision can be used for both procedures.  You can reduce both the fullness of each ala as well as the distance between them.  -Dr. Rolfes

You can combine a lip lift with an alarplasty, specifically a sill reduction. I would not recommend wedge excisions along with a lip lift since the alarfacial sulcus gets violated and webbing can occur. As for the Italian lip lifts they are not recommended because of paradoxically worse scarring. Gary Linkov, MD
Lip Specialist / RealSelf Lip Lift Medical Reviewer
Manhattan

Though I am not familiar with the term Italian lip lift, you can certainly have a lip lift and alarplasty at the same time.  The nostril incision are similar for both procedures however, in order to narrow the nostril the incision needs to be deeper on the inside of the nostrils than what you would typically do for a lift lip alone.  

I see many patients with flaring nostrils on whom I do alar plasty combined with a lip lift. In some cases I design the lip lift incision in such a way to reduce the nostril floors and reduce the flare. Look at my before and after pictures and you will see cases where the alar flare has been reduced. There are however patients on whom I would have reduced the alar flare but they did not want it done.Therefore the answer to your question is a big yes. You would obviously need to be seen in person or at least on video to determine if this is a good option for you.