If its on the way out then grub it out, but also dig out a large area of the soil and refresh with some good decent soil and enrich with well rotted compost, you may like to plant with elaeagnus the one which has the lovely deep yellow leaf margins.

Mixing in some grit, and some lime or gypsum--depending on soil pH--will also help.

first why do you think it has root rot?
does this area get prolonged periods of standing water?

if the roots are rotting you need to identify the cause. Was there any fungal growth associated with the shrub last autumn or during the winter months? personally I'd be wary about planting another shrub in the same area straight away if you don't know the cause of the problem. Perhaps leave the space fallow for a season, place an ornamental pot of summer bulbs or bedding in the place for this year, filling the hole so to speak.

welcome to GoY too :o)

As it's an Aucuba and looks as though its on its way out, I would guess that the leaves are blackening and then dying so it might well be a variety of Phytophthora. If you do a Google search you will find that a few shrubs are immune. Firstly, I would try to remove some of the soil from the hole, fill with new soil and perhaps leave for some time if possible. If you feel that you must replace with something else, then consider using Rootgrow in the hole before planting.