If I watch only one minute in the middle of an 1-hour long YouTube video could this save bandwidth?
Given that the video is not entirely loaded during this one minute of course.
You can't limit the bandwidth used to only that minute's worth of video, because YouTube is going to keep streaming more of the video into your buffer, which will use more bandwidth. Skipping to the middle will obviously skip over the first half of the data, but if you watch a minute and then just sit on the page, it will keep trying to load more.
I don't know if you sat there long enough if it would start trying to backfill the video you skipped, but it will definitely try to buffer the video from where you are at until the end. The only way to guarantee that it will stop using more bandwidth is to leave the page.
One other tip for saving bandwidth on YouTube is that you can jump right to a specific time by adding some info into the URL string. If you want to start at 20 minutes and 45 seconds into the video, add the following to the end of the URL in the address bar: