If the switch interface is set to a single VLAN (access mode), then it really does not matter what VLAN is set on the port, it will be untagged. It is exactly the same for the AP.
If VLAN 1 doesn't work, but VLAN 20 does, then the VLAN1 is not setup properly on the switch, or DHCP doesn't work on it, etc. Again, test it with a laptop.
If the port is set as trunk (802.1q), then multiple VLANs are set on the interface, usually one of them as default, or untagged, usually VLAN 1. When you plug a laptop into it, the laptop will use untagged traffic, VLAN 1. If it works, then the switch is good. The AP has a setting for what VLAN to use for management and what to use for different SSIDs. The management has to be set to VLAN 1. You can verify, by plugging the AP directly into a laptop nic. If you are running Windows, turn on Internet sharing on its WIFI card. This will turn on a DHC{ server in the laptop and the AP will be offered an IP. Verify that you can reach the AP. If yes, then your AP is set correctly to use untagged traffic for management.
But from what you are saying, it looks like the switch port is set to access mode with a single VLAN on it. The AP has no way of knowing what VLAN is set on the switch interface, since the traffic is untagged.