I don't have access to the Internet here. --> Good
Don't you know someone to borrow from? --> Good. Could be improved w/ "Don't you know someone from whom to borrow?" You'll often hear native English speakers saying it like you had written it first; however, it is grammatically incorrect to end a sentence with a preposition such as "from." Again, this is a mistake that many native English speakers make when talking, so it's likely that no one will notice. But in writing, this stands out a bit more and it is best to try and not end sentences with a preposition.
Don't you have someone to borrow from? --> Good. Could be improved w/ "Don't you have someone from whom to borrow?" See explanation above.
I will always think appreciatively of you. --> Consider improving the sentence in this way: "I will always appreciate you."
I will always remember appreciatively of you. --> Incorrect. You cannot remember someone appreciatively. You can remember someone or you can appreciate them; alternatively, you can appreciate someone when you think of them. This would look something like: "When I think of you, I am grateful," or, "I will always be grateful for you." Grateful means "to show an appreciation of kindness." It's a good word to use!