Translation:I eat outside the restaurant.
I think "outside of" and "outside" are both grammatically correct, although the former is not common in some regions. "outside at" is a slightly different meaning, I would interpret it as "out of doors, near the restaurant". Since someone said in the Japanese it means "in the outside area of the restaurant", perhaps they're being fussy. Both "outside of" and "outside" are accepted.
"レストラン" is the Japanese word for restaurant, it's just a loan word, called 外来語 (gairaigo). In Japanese, loan words are indicated by its use of katakana. Other loan words in Japanese are パソ (pan (bread in spanish)), フライト (flight), コーヒー (coffee) etc. And hey! English has loan words from Japanese (and other languages) too! Think tsunami, zen, objects from Japanese culture like sushi, karate, origami, etc.
I'm not sure about a word for restaurant in Japanese before the English origin word was introduced but language changes overtime so even if there was, レストラン is likely the word you'd hear most often in Japanese as of now.
I think this may be more naturally translated to English as "I will be eating outside the restaurant"? I'm not sure because English speaks of it as "where you're sitting WHILE eating" and in English that'd be "I will sit outside to eat", almost. It's rarely phrased quite that way; at least in the states, I've only ever heard it as a seating context. "Would you like outside or inside seating?" can be responded to with "I'll sit outside." and "I'll eat outside.", but by itself, "I eat outside the restaurant" doesn't come across as natural English.
Can't think. No caffeine yet. -_-
Really, you should translate this sentence as "I eat outside (of) the restaurant." If you're translating a text or something from Japanese into English, you should use the most common phrases. For example Translating "食べ物が欲しいです" as "Food is wanted" is technically correct but not standard in English for when expressing that you want something, so when translating, you would instead say "I want food", to make it sound more natural.