What is the meaning of "there goes my day"? My friend message me with a youtube video followed by "there goes my day"
There goes my day means that something has just come up that will occupy that person for the rest of the day, and they won't be able to get anything else done. It's always used as a negative, that is the person would rather be doing something other than what they are having to do.
It's an exclamation of disappointment in this state of affairs.
In the case of your friend, I suspect it is being used humorously - the video is so entertaining that they'll have to keep watching it all day and won't get anything else done.
It means he is going to spend all the rest of his day watching that videos and others related. Generally, the expression "There goes..." means it has come to an end, in this case because he is not going to get anything done, instead he is going to watch YouTube videos all day. With YouTube videos, of course it is to be intended figuratively and not literally.
You can look at some examples with other objects and situations in this WordReference thread.