I have a pinus mugo planted in a tub but half of it is turning brown. Can it be saved?

First of all, is it the straight species, or is it a cultivar ?
Also, is it actually turning brown or yellow ? as there are cultivated forms of mugo that do start to turn either yellow or golden at this time of year, and hold onto that colour until new growth begins next spring.

Just how sunny a position is it in? And, if you planted it this year have you watered well? Whilst you can grow conifers in containers it takes a little more effort,

Not being picky, just going for clarification. When you say half the tree is turning brown, do you mean a large patch, or just half the needles spread over the entire tree?

If whole branches are going brown, I would cut them off. Give it extra care and attention next year and you could be surprised.

If there is an overall mass of brown needles, don't worry. Although a conifer, even these loose old needles.

Pines are very difficult to transplant successfully. It's essential to place some of the old compost in with the new as the old compost contains essential mycorrysa (sorry 'bout the spellin') that the pine needs to feed.
Do not bare roots and collect the biggest rootball you can cope with retaining as much root as possible.
When root pruning is required (every 3 - 4 years) only lightly trim 1/3 of the roots that year, 1/3 the following year and the last third the following year. Grow in full sun. Do not overwater. If all the needles on a branch are brown, then the branch is dead and can be removed. Leave a stub, don't cut it right back to the trunk.