Currently trying to log in into Xbox Live on my 360, however each time I try it fails, keeps saying connection blocked, My home internet system was recently upgraded so I'm thinking that could be the issue, but I'm hoping I can explore the 360 marketplace before the closure in July.
I'm not sure if currently supposed to fixed, however Xbox Live on the 360 has had various problems recently: https://www.trueachievements.com/news/xbox-live-outage-xbox-360-backwards-compatible I had issues connecting last year when doing the same as you wish to.