Apple - Can I store all the content of my previous iPhone in a new iPhone using iTunes?

Yes.

The easiest way to do this is to do a full backup of your iPhone 4s via iTunes.

Then you would connect the new iPhone to iTunes as well and then do a Restore from the iPhone 4s backup to your new iPhone.

Important

The iPhone 4s was available in various sizes, namely 8GB, 16GB, 32GB and 64GB. So you will need to ensure that your new iPhone has as a minimum the same capacity as your old one.

You should also select the "Encrypt local backup" option as this will "allow account passwords, Health and HomeKit data to be backed up". Of course, in your specific case, an iPhone 4S doesn't have health and home data.

I would do it through iTunes using the USB cable. iClould is all well and good if you have more storage on there than the paltry 5Gb Apple gives you as standard. I was never able to backup fully everything from my phone to iCloud as it was over 15Gb including all the photos and videos. Doing it using iTunes is simple and just takes a little time is all.

Just a little FYI... There are some apps that won't be restored from a full backup. I owned the app 'Music D/L' which let you download MP3 files from a website and listen to them on the phone. Since it has been removed from the app store, you don't get it back on a restore. I'm not sure if this applies to every app that gets removed or deleted from the app store.