Sightseers should book Dubai in January. Sunbathers who want real heat should pick another month. This is the coolest month of the year, with highs around 24°C and lows near 14°C, so you can be out on foot at noon, which for much of the year is out of the question. There is sun for close to ten hours a day and rain on only about two days, so plans rarely need moving.
Book the desert first. Put it on your second or third day, which leaves room to rebook if the trip is cancelled. For empty dunes instead of a crowded camp, pick a trip into the Dubai Desert Conservation Reserve. You cannot drive in yourself, and only a small number of approved operators are allowed inside, so check that yours is one of them before you pay.
Take a warm layer that evening. You will be sitting still on sand after sunset, and that is the only time in January when Dubai feels properly cold.
Give the Creek a full day and do it in this order. Start at Al Shindagha Museum, near the mouth of the Creek, where the restored houses along the water explain how Dubai grew up around it; allow the morning. Walk on through the textile souk in Bur Dubai to the Al Fahidi Historical Neighbourhood, with its wind towers and courtyard houses. Then take an abra across the water to Deira for the Spice Souk and the Gold Souk. The lanes cost nothing to wander, and the crossing costs next to nothing.
Keep one day for Hatta, in the Hajar Mountains near the Oman border. You do not need a car. The Hatta Express bus runs from the Dubai Mall bus station and takes about 90 minutes.
Hatta Wadi Hub grades its trails green, blue, red and black. Green is gentle enough for most people and climbs to the giant HATTA letters above the dam. Black means long, steep climbs over loose rock, so do not make it your first hike in the region. Wear closed shoes, and check the times back before you set off: buses run only every couple of hours, and the last one leaves Hatta in the early evening.