The waiter soon bought him a cup of tea.
= The waiter soon bought a cup of tea for him.
即：buy sb sth = buy sth for sb 给某人买某物
不过,题目应该是brought,句意才通顺,此时为：
The waiter brought him a cup of tea.
= The waiter brought a cup of tea for/to him.
即：bring sb sth = bring sth to/for sb 把某物带给某人/为某人带来某物