Automatically update multiple file names?
There are a number of ways you could do this. Below I explain two possibilities.

Do this natively within the Finder

You don't specify what version of macOS you're running. But a feature that was first introduced in Mac OS X Yosemite will do exactly what you need. To use this feature you would use the following basic steps:


Copy the folder named "41777 Co. Name – (Business Cards)" and rename it to "52888 Co. Name – (Business Cards)"
Open the "52888 Co. Name – (Business Cards)" folder
Select all five files within the folder
Right-click on the files
Select the Rename 5 items... option from the context menu that appears (obviously, the '5' will be whatever number of files you selected at Step 3)
Select the Replace Text option from the drop-down menu in the window that appears
In the Find field, enter 41777
In the Replace field enter 52888
Click on Rename


Note: You may want to also check out the Add Text and Format options to explore the other possibilities.

A Better Finder Rename

If you're looking for much more advanced options, or are still using an older version of macOS, then A Better Finder Rename is an option worthy of consideration.

This software provides a huge range of renaming options - too many to cover here - but it's been a well supported package for at least ten years now. You can also download a free trial (assuming you have Mac OS X Lion or above).

Other native options

Mac OS also provides AppleScript, Automator and Terminal, all of which could be used to batch rename files in the manner you want. Since it's most likely that you'll be able to use the Finder option, and because using AppleScript, Automator and/or Terminal is a bit more involved, I won't go into any detail here.

However, should you wish to have more info on these options instead, let me know and I'll update my answer.