Upon landing, you’ll be within Lanayru Mine, rather than the desert itself.
Goddess Cube 19
Anyhow, go down to the lower level. Go around to the back of the tower to find Goddess Cube 19; hit it with a Skyward Strike.
Restore your Bomb supply back here as you need to, then go north towards the Bird Statue. Use it and go along the path into the cavern.
Inside, defeat the Keese and push the cart until it falls off of the track. Defeat the Yellow Chuchus nearby and push the cart to the wall. Dash and jump up the cart and the ledge.
Continue along the path and around the corner. As you continue, go into the alcove on the right side to find some Yellow Chuchus. Defeat them and push the mine cart to the end of the track. Go into outside area ahead.
You’ll find a purple crystal – a Timeshift Stone – here. Striking one will transform the nearby area to how it was a millennium ago, back when this place was more technologically advanced.
Use your Beetle and grab the Rupees from the surrounding conveyor belts and hit the Timeshift Stone to time travel back a thousand years. Enter the mine cart in this area and you’ll head to a northern room. Inside is a chest with a Red Rupee.
Get back out of the room (get back in the cart), then go east and into the cart you pushed most recently. You’ll ride to a new southern area.
Anyhow, Bomb the rocks around here to find another Timeshift Stone. Hit it and continue forward. Throw a Bomb into the bowl-thing the statue is holding to make it fall, revealing a room.
Go through to find quicksand. Quicksand, of course, sinks you – you must dash across it. Whenever possible, try to have full stamina prior to traversing quicksand.
Anyhow, Bomb the statues’ bowls as you go across the quicksand. Be sure to investigate the tunnel behind one of them - there is a chest in there with a Red Rupee inside.
Continue past the statue section. Defeat the Keese in here and continue forward some more to a huge room.
Here, you’ll find Electro Spumes. They can be annoying, but a Bomb will finish one off. Go across the quicksand, by going from ledge to ledge, fully restoring your stamina prior to going to the next ledge.
Go to the northern end of the room, where you’ll find a mine cart. Push or pull it along the track as much as possible. Continue uphill.
Defeat the Keese here and take the western doorway. You’ll find a long “alleyway” of quicksand and a lone Bomb Flower. Restock some Bombs, then begin rolling Bombs across the quicksand to destroy the rocks.
In the room at the other end of the quicksand, there is a Timeshift Stone beneath one of the rocks you destroy. Once you find it, send your Beetle across to hit it. Once you do, you can now walk across the grass to the other side.
Continue along the path to reach a chest in the large room - it contains a random rare treasure. Go down, to the ground, and get into the mine cart. Since it is now activated, you can ride north and out of the mine.