You should change your tires for safety sake of course. I DO have the original tires on my CA160 and they are a little cracked, yes, and a little worn down. I ride super conservative on that bike making sure to baby it while making turns. This is a good practice even with good tires because your not going to get anywhere any faster with your 6-9hp and crashing an old classic Honda is not an option.
I have spent a lot of time researching ans searching for classic bike tires. The absolute BEST place to order up some tires?
JC Whitney.com ! Go do a search for you tire size 3.00 and have a look! They have an awesome selection of tires. Now MAKE SURE that you buy some inner tubes to replace your old ones as well. I bought the last tubes that were in stock at my local bike shop as I was shocked they had them in the first place. You might want to buy an extra tube or two just in case you get a flat in the future. They are so cheap it's not even worth your time to patch them!
To make the tire swap painless buy some nice tire irons. Now that tools are so cheap and readily availible there is no excuse not to buy some nice ones. These are really easy tires to change and leaving them in the sun for a while before you take them off makes the change even easier. There should be a rubber band that goes inside the tire that protects the inner tube from being pinched or punctured by the spoke nuts that come into the rim on the inside. Just use some duct tape of some type to cover these up if your bands are in bad shape.
Not that you don't know all this stuff but it's good to post this stuff for people to read.
Let us know how it goes and which tires you get. You can email me pictures and I can post one or more for you if needed.