Custom encoding for BinaryReader
BinaryReader.ReadString() expects the string prefixed with the number of bytes to read, not the number of characters (I think this is because of variable-length encodings, especially UTF-8, but also UTF-16).

So, you can't use ReadString() directly, but you also don't have to convert the characters byte by byte like you do (which wouldn't work for non-ASCII characters anyway).

For this, you can use ReadChars(), which takes as a parameter the number of characters (not bytes) to read.

You also need to figure out what format is the number of bytes saved in. It could be a simple single-byte number (which means the string can have at most 255 characters), or it could be VLQ-encoded, which you can read using Read7BitEncodedInt(). Though that method is protected, so I'm going to assume the former for simplicity.

So, the code could look like this:

using (var reader = new BinaryReader(File.Open(fileName, FileMode.Open), Encoding.Unicode))
{
    int characterCount = reader.ReadByte();
    char[] characters = reader.ReadChars(characterCount);
    return new string(characters);
}