Someone who is annoyed at the coverage of her scandalous story in a news broadcasting tv programme (like bbc or cnn) is having a argument with director of the news network channel as to why she run the story.she says" it is sitting very high on her(the director) half hour".
"Sit high" is a metaphor that derives from several sources. In one sense it can simply refer to physical position ("The house sits high on a hill"), but it can also (presumably by analogy to a king's throne, etc) refer to one's social position or position of authority.
But in the referenced piece it sounds like "high" simply refers to the position of the piece in the broadcast schedule or list of priorities.
sit = "to be in a particular position"
"The town sits at/in the bottom of a valley".
The verb is used in the same way to describe the position of items in an agenda. Those that sit high are those near the top of the agenda. Because a program has a similar list (schedule) of items to be considered during the broadcast, some items may be described as sitting high on the schedule (for "the half hour" of broadcast). The implication is sometimes that those items considered first in any such list (agenda or program schedule) are the most important.