Plants have adapted to live in the arctic, however few have the looks of cacti. The answer might be that they could, but other shapes and forms are better at the filling this niche. The vegetation in the arctic are mostly low growing herbs and grasses. The only wooded plants are small shrubs.
The problem here is rather to much water. While the arctic is dry, it's also freezing. The cold air can't hold much moisture so the water only evaporates very slowly. This is why tundra regions have bogs despite very little precipitation. But much of the time the water is frozen however and inaccesable to plants. Cacti have a upper hand then but the strategys already used (shedding leaves, survive underground) is probably better. There is no use to stand out of the snow in the freezing wind when there is no sun anyway.
Cactai have a advantage in the desert in that they can store their water and use it to make food while others can not because the soil is dry. 778
In the arctic water is available at the same time as the sun, more or less. In summary, storing water in the arctic is probably not nessesary.
On a other world where the tundra reaches down to the tropics there might be a different story, then the winters would be cold and sunny.
However... cactai might be better adapted for holding on to heat... Let's run with this idea and design a extreme cacti able to live even in Antarctica!
Why would a cacti be able to survive beond the tundra, where no vascular plant have ever grown before?
They are roundish, less surface area equal less heat loss. The same adaptations make them retain water.
They have hairs/spikes to insulate what little heat they get. Just make them transparent to let the sun trough and make the cacti dark to absorb the sunlight.
Hmm, not enough for Antarctica? Well then:
They have to be small so the snow can cover them during winter. Yet large enough to hold on to heat.
Have them run sun-warmed fluid down to the roots, or better yet, absorb the water and nutrients above ground and get along without roots. They won't need much anyway since they are growing so slowly. If they use this strategy they only need the snow to melt occasionally, the ground can stay frozen.
Young plants will have a harder time staying warm, so they might start to grow inside the parent to then be ejected when large enough. They then disperse by rolling in the wind, a lucky few will find a sun-facing rock to grow against.
Still not enough? Well, we can always resort to magic. 591