Once you've setup the Slingbox for viewing locally, its the routers responsibility to allow remote viewing. I recommend that check your router's firewall rules. You'll need to make sure that that the corresponding public UDP and TCP ports and being forwarded to your Slingbox's private IP address. Most of the time this problem is associated by only having the TCP port forwarded.
Sometimes, you have to deal with either the broadband provider or WiFi Hotspot either blocking the ports, throttling the bandwidth, or using them for their own nefarious purposes. You can try to work around this by changing Slingbox's default service port of 5001 to something more common like port 80 (HTTP) or 443 (HTTPS). If this is necessary, then you also need to change the port forwarding rules to match the new port number.
IP address and Port Number: http://support.slingbox.com/get/KB-005866.html
Port Forwarding: http://support.slingbox.com/get/KB-2000068.html
Network Setup Assistant: http://support.slingbox.com/get/network-assistant.html
Manual Changes: http://support.slingbox.com/get/KB-5000126.html