About how much should it cost to upgrade my 1970's 150 amp circuit box to a 200 amp box??

Before you take that step, Why are you moving up your service?

Depends on the market your located in. Boston will be more expensive than Fort Kent Maine. And depends on the lengths and types of materials used or required, such as copper or aluminum wire, PVC or steel conduit, overhead or underground, etc.
etc. Permits and utility charges vary also.

Anywheres from $1K to $2K for budget purposes.

Find 3-4 small to mid size contractors, ask for estimates, be sure they do service size calcs to size the service, ask for as much detail for scope of work as possible, require each to apply for permits and inspections if required, and ask for certificate of insurance to confirm liability and worker comp insurances.

If the Runs were all romex, would be extremely more faster and cheaper.
The more solid conduits, the worse, If all the pipe runs coming from the top, or coming in from the bottom would be good.