How to solve this problem by pointers

This is your homework, so we aren't going to do it for you - that would defeat the point!
It is there so that you think about what you have been told, and try to understand it. It is also there so that your tutor can identify areas where you are weak, and focus more attention on remedial action.

And that code isn't very good, mostly because you haven't tried to do what the question asks you to.
Start at the beginning: read the question carefully. The first thing it says is that users should be able to add new friends. The second is that it's "as many as they want" - so your fixed size array is not going to work, even if you did give it a size, which you haven't.

This needs pointers, as you said in the title. So create a struct for each friend, and write a function which creates a friend (using malloc) and lets the user fill in the details. The function then returns the friend and the main function then adds it to a list.
Get the function working first, creating a freind structure and populating it with user input. Then write a function to print information from a friend structure. Test them both thoroughly. Then start working on the list part.
Break it all down into smaller pieces and it should be a lot easier to do.

I have no intention of solving this as it homework you need to learn it. I will however give you a good start point

// FORWARD DECLARATION
typedef FRIEND* PFRIEND;

// RECORD THAT DESCRIBES EACH CONVERSATION ... HINT SINGLE LINKED LIST
typedef struct ConversationRecord {
	PFRIEND party1;									// Party talking to
	long talkTimeInSeconds;							// Conversation length
	long daysSinceTalk;								// days since spoken too
	struct ConversationRecord* nextConversation;	// Ptr to next conversation
} CONVERSATIONS;

// RECORD THAT DESCRIBES EACH FRIEND .. HINT SINGLE LINKED LIST
typedef struct friendRecord {
	char* personsName;								// Persons name
	CONVERSATIONS* lastConversation;				// Last conversation record
	PFRIEND nextFriend;								// Pointer to next friend
} FRIEND;

// Global list of all friends
PFRIEND friendList = 0;

Now start by creating the list of friends into friendlist.

It's all single linked lists, you may later decide to change to double linked list for speed. Now read up on lists and do the homework.

Quote:

Users should be able to add new friends (as many as they want!) and store the number of days ago that they last talked to each friend. Let users update this value (but don't let them put in bogus numbers like negative values).

If the data is to be stored in a database, it is a bad idea to store the number of days since last talk, because tomorrow they all will be wrong.
The right solution is to store the date of last talk instead.
So intead of storing user input, just subtract from current date and store the result which is date of last talk.