HI Richard,
I think you may have stumbled into a bit of a messy situation here. If you do not have access to the old site, how can you 301 redirect to the new site? The 301 redirects are put into the .htaccess file of the old site. Think of it like a "We Moved Sign" you would place on a retail store. You pull into the parking lot and see a sign that says. "hey we moved, we are now located at...". If you're not allowed to use the parking lot, you have no where to put this sign.
You need to make sure your client is allowed to, or at least the company hosting the site is willing to allow you to put in the redirects. To answer your question you need to specifically put in a the .htaccess file on the OLD site to the new site.
You do not have to be concerned with matching URL's only that you format the .htacess file correctly. An example would be (again this must be on the old site server).
If you choose to keep the same file structure you can do a blanket 301 redirect using mod rewrite (it must be enabled on the old server). Like this.
Options +FollowSymLinks
RewriteEngine On
RewriteCond %{HTTP_HOST} ^OldDomain.com$ [NC]
RewriteRule ^(.*)$ http://www.NewDomain.com/$1 [R=301,L]
You can read more about redirects here at Moz.
If you do not have the ability to put an htaccess file or the clients hosting provider is un-willing to help (they may for a charge) then you're in a bad situation with no great solutions. The only thing I can think of is having the clients old host create a page that say this site has moved and provide a link to the new site. Our company did this many many years ago http://www.columbiaindustrial.com/ when they changed their name and host (before my time). Then you need to recommend how long you would recommend the client pays for this new page to be live... I'd say at least 3-6 months.
Hope this helps
Don