Finding a (local) person right away I guess will be difficult but probably depends on the budget. But I don't have any experience with these PC Doctor companies that come in and fix problems, you might want to try contacting a few of them with your exact requirement list. But be careful with your data (I found r_admin installed on a Windows Server after one of these companies hired by our legal firm came to do "stuff").
I took care for years myself or one of my partners did, whoever was around. But after a certain size we hired staff (ie. customer support email reply person) that also had computer knowledge and we preferred people that grew up overseas but still with HKID for those jobs. It always helps if the leadership also knows what they are doing though.
We then started to create knowledge documents "how to setup email" (with email server info etc), "how to create a backup" (where to find software, how to install, location of backup files), etc. The person in trouble and the person troubleshooting the PC could then rely on them.
Maybe something to keep in mind when you hire in the future. I understand you are in a different field than I was, so you probably don't need technical support engineers or customer support engineers who often come with added computer knowledge, but if you hire locally it never hurts if they have a bit of computer knowledge on the side. Let them setup their computer and let them create these documents. Builds understanding and something for the future. If the first version is not perfect, let somebody else who you hire in the future look over them and revise them. Having a bad document is better than having no document.
Another alternative would be for your NZ guy to look into remote admin tools. For most computer problems you don't have to physically be in front a of a PC. This may require a bit of planning and setup (opening up firewall), installing/enabling remote desktop/remote assistance on Windows or other stuff on the Mac, but it works very well. This way you call him to explain the problem, and he then logs in to the computer with the problem to have a look. It's just like if he would be sitting in front of the computer. And since you are moving, somebody will probably have to plan/setup your firewall and internet anyway, so it goes without extra effort. Let me know if you need any (free) "consulting" with this and we can discuss this over a coffee at Starbucks down at the Metroplaza.
It's Kwai Fong. We don't like Fung's around here 
Good luck.