I did this just this past September. I just made sure I had very clear US Custom 4457 forms (in fact prior to travel I politely asked the us customs agent in Chicago to sign/stamp 6 original form 4457 for me so I was presenting an original US document in Zim ), along with my invitation letter, passport, and a copy of my airline itinerary. I also did not prefill the import forms and They had me fill out the Zimbabwe import form as well as the declaration for cash brought in to the country in front of the Zim Customs officers. They checked my firearm serial numbers against the 4457 and counted Ammunition to confirm number of rounds, checked my invitation letter, and checked my passport.
Additionally I used Wengayi (wenhau ? Not sure spelling) service in and out of Zimbabwe. Primarily appeared to expedite the lines at customs and avoid some scrutiny by local airport staff/officials. As a first timer I appreciated the help and do not believe it was a significant cost.
My advice is be organized, be patient, be polite, and be courteous.