Creating your own strcat is nothing, it takes literally a couple of lines.
void my_strcat(char *dest, char *source) {
// Find null-terminator
while ( *++dest );
// Add new string
while ( *dest++ = *source++ );
}
Making it safe would be the next problem. But think about what Lerner is telling you, you're assuming what the user inputs will be shorter than 128 characters, if you can't use std::strings, try learning how to use dynamic memory, or validating your input.