Anonymous types allow you to select arbitrary fields into data structures that are strongly typed later on in your code:
var cats = listObject
.Select(i => new { i.category_id, i.category_name })
.Distinct()
.OrderByDescending(i => i.category_name)
.ToArray();
Since you (apparently) need to store it for later use, you could use the GroupBy operator:
Data[] cats = listObject
.GroupBy(i => new { i.category_id, i.category_name })
.OrderByDescending(g => g.Key.category_name)
.Select(g => g.First())
.ToArray();