Enumerations (enum)

/**
 * Get the current day's predecessor.
 * 
 * Example Starting from {@link #TUESDAY} we get {@link #MONDAY}
 * 
 * @return The day's predecessor (yesterday)
*/
public Day predecessor() {
  final int ordinal = ordinal();
 final int ordinalOfPredecessor = (ordinal + values().length - 1) % values().length;

 return values()[ordinalOfPredecessor];
}

The desired output contains both a given direction's oral description and a 0 - 360° degree value. We thus start by:

public enum Direction {
  int degree;
  String fullName;
}

For creating a Direction instance a constructor accepting both values is being required:

public enum Direction {

  N(0, "north"); ❶

  Direction(final int degree, final String fullName) { ❷
    this.degree = degree;
    this.fullName = fullName;
  }

  public final int degree; ❸
  public final String fullName;
}

For creating output texts like e.g. north by west (315°) we need a toString() method. Also adding the yet missing directions our final result looks like:

public enum Direction {

  N(   0, "north"),
  NE( 45, "north by east"),
  E(  90, "east"),
  SE(135, "south by east"),
  S( 180, "south"),
  SW(225, "south by west"),
  W( 270, "west"),
  NW(315, "north by west");

  Direction(final int degree, final String fullName) {
    this.degree = degree;
    this.fullName = fullName;
  }

  @Override
  public String toString() {
    return fullName + " (" + degree + "°)";
  }

  public final int degree;
  public final String fullName;
}

Using a switch allows for a straightforward and simple solution:

public enum Direction {
  ...
  public Direction opposite() {
    return switch (this) {
      case N  -> S;
      case NE -> SW;
      case E  -> W;
      case SE -> NW;
      case S  -> N;
      case SW -> NE;
      case W  -> E;
      case NW -> SE;
    };
  }
  ...
}

Following the provided hint we may instead calculate the opposing direction:

public Direction opposite() {
  return values()[ (ordinal() + 4) % 8];
}