You already have a great answer, but to clarify the points you asked about, 外国でも should be interpreted as
外国で + も
where 外国で使える means "able to use in a foreign country", i.e. "works abroad". も is "also", so that 外国でも使える means "also works abroad".
As I wrote in my comment, ～てきた is used both metaphorically and literally. In this case, it is used literally and 借りて来た means "I borrowed and came", i.e. "borrowed before I came", or here
(Before I came,) I rented a mobile phone, which also works abroad, at Narita Airport, so there's no problem.
The English translation is a bit cumbersome, but such is the nature of literal translations.
Edit. A different translation (partly due to snailboat) could be
I went and got a rental phone which also works abroad at Narita Airport, so there is no problem.