Learn Finnish: Asiakaspalvelu hoitaa postikulut puhelimitse. - Customer service handles shipping costs by phone.

The prolative ending -tse is almost always added to the plural stem of a word, even when the meaning is singular. For puhelin (phone), the singular stem is puhelime- and the plural stem is puhelimi-. Add -tse to the plural stem, and you get puhelimitse.

Since both can mean by phone, what is the difference between puhelimitse and puhelimella?

You can use the adessive case puhelimella to mean the exact same thing, and it is much more common in everyday speech. The prolative form puhelimitse sounds more formal, administrative, or official, which perfectly fits this sentence about customer service policies.

Why is the object postikulut in the nominative plural form rather than the partitive plural postikuluja?

Because the customer service takes care of the entire shipping cost, making it a total object. When a total object is plural, it takes the nominative plural form ending in -t. If they were only paying for a portion of the costs, you would use the partitive plural postikuluja.

I have seen the verb hoitaa used in medical contexts. How does its meaning change here?

While hoitaa does mean to treat or to care for a patient, in business and everyday contexts it translates to handling, managing, or taking care of a task. Here, they are taking care of the shipping costs.