I think the only area that you will need to be careful is with IP addressing.
Assuming that the existing 4-port switch is assigning IP addresses via DHCP, you will need to ensure that the WAP DHCP scope does not overlap it.
Alternatively, you could switch off DHCP allocation from the Linksys as long as it is able to forward DHCP requests to the WAP.
Of course it is possible to turn off DHCP and use fixed addresses for all nodes.
With care you can mix modes - all hard-wired devices have fixed addresses, with only WIFI nodes using DHCP. Again you just have to be careful that the addressing scheme is consistent and that fixed & DHCP don't overlap.
As to what type of cable you will need - it depends on whether the WAP and / or Linksys have uplink / downlink ports or a switch port (MD / MDX). If they have, then these effectively act as the cross-over and you actually need a straight through cable. to cascade a hub or switch using a standard port does indeed require a X-over cable.
As long as you note down your existing configuration, you should have no problem experimenting - you can always restore things as they were.
Oh yes - and RTFM
