Shifting from Dubai to Riyadh

All of that is something that your husband's employer needs to help with.

But basically, how it works is (in simple terms):

He gets a work visa
Enters Saudi and does his medical
Employer applies for an Iqama / residence card for him
He gets Iqama
He then sponsors you as a dependent (if his profession on the visa allows for it)
You then enter and go through the same process

I would not recommend a visit visa as Saudi is not like UAE i.e. there isn't any conversion of a visit visa into a residence or work visa.

The timings vary depending on:

1) How strong your husband's company HR are in the government processes
2) Are all your necessary documents attested?   (by Saudi culture and Saudi Embassy in home country)

If the answer to the above two points is positive, then it is possible to get you into Saudi within a month.   But be aware that generally it takes longer.