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You keep your router (with its internal network switch). This makes you an internal LAN, separate from the WAN/internet.
I want to send this POS router back and buy one that does both tasks, rather than buying two things. This is probably related to my comment to snotty about being cheap.
If you want to completely open one computer to the world (are you really sure?)
When I say open to the world, I mean have specific ports open to the world to do the things I want to be able to do (web and ssh traffic forwarded to a specific computer).
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Aha. You're currently renting the Zyxel/have bought it and want to return it? I misunderstood.
In that case your suggestion of buying a router-switch-AP (these are the many jobs that one Zyxel box was doing) is fine. Pick your budget and spend as much as possible or buy a cheap router-switch and buy a separate AP, either option works. You need to adjust the DHCP settings (to set your server a reserved IP address) and port forwarding settings in the router to forward the right ports (80 and 22 for HTTP and ssh respectively) to your reserved IP. Best to set up a dynamic DNS (DDNS) service to keep track of your external IP as it might change. I use no-ip and it works fine.
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The cost of a good router isn't much different to a Unifi WiFi AP and their Edgemax router which offers a great deal of flexibility in terms of port forwarding. It's a less neat solution than an all-in-one router and wi-fi point but more flexible and more future-proof.
If you want to look at the cheaper end of the scale then Edimax seem to be decent for the price.
hamrack
aggi
It will be perfect, you just need to set it up right.
You keep your router (with its internal network switch). This makes you an internal LAN, separate from the WAN/internet. You use the AP, plugged into the router, to set up a WLAN that allows your to talk to the LAN over WiFi (with sweet speeds). Turn off the WiFi on your Zyxel to avoid conflict.
If you want to completely open one computer to the world (are you really sure?) then you need to set up the router to do this for you - it's the gatehouse between the external WAN and internal LAN. You can either put that computer into the DMZ (yes that means de-militarized zone, it's a gaming feature most routers have) and it will appear completely open - this is dangerous - or you can selectively forward ports on the router's firewall to your internal device. I would suggest the latter.