I am ashamed, sir, that I wasted the many hours () I should have studied in class.

I should have studied in class.这个从句缺时间状语,所以用when引导

when 表示特定时间段

I am ashamed, sir, that I wasted the many hours () I should have studied in class.

解析：
我很羞愧,sir，我浪费那些我本该在教室学习的时间。
这是定语从句。先行词hour，时间，引导词就用when。就这个表示时间。

从另外一个角度看。I should have studied in class.我本该在教室学习的（..）所以也可以得出结论是when。