Great questions and I am glad to hear that you are taking the long view in your preparation for Everest.
Several things need to be considered in determining if climbing Pumori 4-5 months before you leave for Everset is in your best interests.
Exposure to altitude (even months ago) seems to make subsequent climbs to similar altitudes a bit easier. This is a general rule and depends on the individuals response to altitude. Some folks just acclimate poorly. Our personal experiences is that each trip to high elevation get’s easier for us.
Several of the alpinists we have worked with have used short periods at altitude as training camps only a couple of months before major expeditions. Read these articles for more info
This program continues to be one we uses with high level climbers.
The thing to keep in mind is that these “training camps” were less than 2 weeks long and the athletes were very fit before. That meant that as soon as they returned home that could begin training hard again. However……If you plan on a 4 week Pumori climb you will will very likely be quite fatigued and depleted when you get home and have lost a fair bit of your fitness base.
I hope this helps