It seems that there is an issue with traffic congestion here.
Try connecting at different times of the day to see if that helps.
Also it wouldn't hurt to upgrade your internet package if you can afford it.
About the NAT:
The Xbox should run just fine without NAT being open, but it wouldn't hurt to open them.
Depending on the router there are different ways of setting it up, but if you have default settings, you should be able to go into the router and change some settings to make it run more efficiently.
Basing off that router that you suggested the issue is with,
Assuming you have default settings, go to 192.168.1.1 on a computer connected through the router (Through one of the ethernet ports) and login with either "1234" as the username and "admin" as the password (no quotes) or a blank username and password "admin"
Then follow these steps on this website, but do not change port 80.
http://portforward.com/english/routers/port_forwarding/Linksys/BEFDSR41W/Xbox_Live_360.htm
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