You could run a trace, this will track what is occurring on the database around the time the user is reporting a performance issues.

However, for the CPU to peak at 100% around the same time each week and there isn't a SQL agent job scheduled then this doesn't sound like a SQL issue (unless there is a rogue user!). By the sounds of things it could be an infrastructure issue. I would ask your Infrastructure team to identify what type of updates occur around this time and request to be excluded from them. If that's not fruitful look into the logs - [start][administrative tools][event viewer]