Does Airport Extreme handle 1GB internet speed?

Airport Express have 100mbit ethernet port.. so regardless of the wifi speed the max throughput is never greater than around 95mbps.

The Extreme does have gigabit but is no where near fast enough as a NAT router.

As per Tesserax comment.. you will never see gigabit over wireless with current products.. maybe 500mbps on a good day with client device in patch cord distance to an AC router if said client has full three stream wireless card.

A number of things to go over here.

1. You are never going to see 1 Gbps throughput through any Apple AirPort router, especially if it is configured as a router with NAT enabled. You may get closer to this if it is reconfigured as a bridge (NAT & DHCP services disabled).
2. If your goal is to use AirPort Express base station to extend the Wi-Fi network provided by the Extreme using a wireless connection between them, expect that throughput performance to half again.

The bottom line? If you want near 1 Gbps throughput throughout you network, use Ethernet connections wherever possible. Wireless will always provide lower performance.