Various simplifications:

void updateWhenIdle() {
   if( spriteIdleCounter > 9 ) {
      if( spriteIdleNumber == 5 ) {
         spriteIdleNumber = 3;
      }
      else {
         spriteIdleNumber ++;
      }
   }
}

  // I have done it in this way, so that it is clear what it should contain,
  // but the ideal, it would be that you do not believe variables, and then
  // add them to the array, but directly instantles the item's of the same
Image images[][] = new Image[][] {
   { up1, up2, up3, up4, up5 },
   { down1, down2, down3, down4, down5 },
   { left1, left2, left3, left4, left5 },
   { right1, right2, right3, right4, right5 },

};

Image image = new Image();

public void draw( Graphics2D g2 ) {
   g2.drawImage( right1, x, y, gp.tileSize, gp.tileSize, null );
   BufferedImage image = null;

   switch( idle ) {
      case "up":
         select( spriteIdleNumber, 0 );
         break;
      case "down":
         select( spriteIdleNumber, 1 );
         break;
      case "left":
         select( spriteIdleNumber, 2 );
         break;
      case "right":
         select( spriteIdleNumber, 3 );
         break;
   }

   switch( direction ) {
      case "up":
         select2( spriteIdleNumber, 0, 2 );
         break;
      case "down":
         select2( spriteIdleNumber, 0, 4 );
         break;
      case "left":
         select2( spriteIdleNumber, 0, 4 );
         break;
      case "right":
         select2( spriteIdleNumber, 0, 4 );
         break;
   }
}

void select( int spriteNumber, int file ) {
   if( spriteNumber == 3 ) {
      image = images[ file ][ 0 ];
   }
   if( spriteNumber == 4 ) {
      image = images[ file ][ 1 ];;
   }
   if( spriteNumber == 5 ) {
      image = images[ file ][ 2 ];;
   }
}

void select2( int spriteNumber, int file, int firstColumn ) {
   if( spriteNumber == 1 ) {
      image = images[ file ][ firstColumn ];
   }
   if( spriteNumber == 2 ) {
      image = images[ file ][ firstColumn + 1 ];
   }
}