Is that all you have for your network equipment? I've never heard of a Catalyst 2800. The Catalyst line are Cisco's campus/access switches. There is a 2800 series router.
With respect to implementing VLANs, no you do not have to run additional cables to the users if you have the proper access switch providing connectivity. The switch will accepted VLAN tagged frames and then take out the VLAN header before sending them on to the end device. If you only have one Ethernet drop for each desk location, you can set up a double tagged VLAN trunk to the Cisco phone. The phone should have two Ethernet ports on it. One is to accept the incoming connection from the LAN drop or switch. The other is to allow a PC to be plugged into the phone. In essence the phone is acting sort of like a mini switch. It will accept a double VLAN tagged frame, then break out the VoIP component and then send the rest of the data along through the second port where the PC is plugged in. Now I'm not entirely read up on this setup as I've never set this up. I usually have a dedicated drop for VoIP phones or multiple drops in a location which allows me to have only a single VLAN go to a specific device. I also seem to recall an implementation where part of the Ethernet frame remains untagged on a specific VLAN for data and then the tagged portion is for VoIP traffic.
With respect to implementing VLANs, no you do not have to run additional cables to the users if you have the proper access switch providing connectivity. The switch will accepted VLAN tagged frames and then take out the VLAN header before sending them on to the end device. If you only have one Ethernet drop for each desk location, you can set up a double tagged VLAN trunk to the Cisco phone. The phone should have two Ethernet ports on it. One is to accept the incoming connection from the LAN drop or switch. The other is to allow a PC to be plugged into the phone. In essence the phone is acting sort of like a mini switch. It will accept a double VLAN tagged frame, then break out the VoIP component and then send the rest of the data along through the second port where the PC is plugged in. Now I'm not entirely read up on this setup as I've never set this up. I usually have a dedicated drop for VoIP phones or multiple drops in a location which allows me to have only a single VLAN go to a specific device. I also seem to recall an implementation where part of the Ethernet frame remains untagged on a specific VLAN for data and then the tagged portion is for VoIP traffic.