You can use different Apple ID's with different services. Create a new Apple ID for the second iPhone and use that ID for iCloud, while still using the same iTunes/App Store login.
There may be a way to manage how the devices sync (using the same account), but I have been using the above method successfully.
Sharing apps is actually pretty easy.
On your iDevice, just go to Settings > Store and sign into the account you want to download purchased apps from, and then when you go to install them, it will act like it's going to purchase them again, start to download, and then pop up a dialog saying you've already purchased this app, click OK to download again for Free.
I do this on a weekly basis with 2 other accounts and have been doing it for over a year so I know it works. Even on iOS 5.0.
Just be careful to switch to the right accounts when you make your purchases. Since it does not necessarily say that you have already purchased the app, you could look up the wrong app (a HD universal version instead of the normal version, for instance) and end up purchasing it by mistake.
Finally, note that you could sign into your alternate Store account, then go to App Store > Updates > Purchased and see a list of what's available for download on that account, BUT NOTICE that this will LOCK your device to THAT account's past Purchases for the next 90 days!
See: http://support.apple.com/kb/HT4627
"When you turn on Automatic Downloads or download past purchases on an iOS device or computer, your device or computer will be associated with your Apple ID. Your Apple ID can have up to 10 devices and computers (combined) associated with it. Each computer must also be authorized using the same Apple ID. Once a device or computer is associated with your Apple ID, you cannot associate that device or computer with another Apple ID for 90 days."
Best Answer
In general, what you are suggesting is a Bad Idea™.
iOS devices such as iPad and iPhone do not have user accounts as you would have on a computer, where each user's data is stored separately and switching between accounts shows your data/apps/photos/etc. versus his. Simply signing in and out of an Apple ID on a device is not only time-consuming (to cover everything, there are at least 4-5 places I can think of off the top of my head where you'd have to log him out and you in, or vice-versa) but it's will wreak havoc with lots of things.
If you use iCloud calendars, they will either be deleted from the device each time you sign out, or if they are left, will be merged with the calendars of the next person who signs in. Notes stored in iCloud would be the same way. Calendars and notes don't necessarily take a huge amount of data, so maybe deleting them each time isn't that big a deal. What does take a lot of data is photos. If you use iCloud Photo Library, signing out will either delete all photos on the device (and therefore have to be re-downloaded each time you sign in), or if left, will be merged with the iCloud Photo Library of the next person to sign in (and eventually you'll end up with identical photo libraries).
If you use other systems for Calendars, Notes, and Email, such as Gmail, your stuff won't be accessible by just signing in and out of an Apple ID - you'd have to remove his account(s) and add yours, or vice-versa. That will cause all such data on the device to be deleted and have to be re-downloaded every time.
Apps do not get deleted by signing in and out, but by the same token, apps don't get downloaded just because you sign in. You can have apps purchased with different Apple IDs on the same device, but to update them, you have to be signed in with the ID that purchased them, which can create a headache. Also, app data, such as your progress in a game, does not go anywhere just by signing in and out. For games which don't have multiple save files (i.e. the majority of them), only one of you can save your progress. For other apps which contains data specific to one of you, the same idea applies.
The specific setting I mentioned in my comments requires signing in and out of iCloud, which I didn't think about when I mentioned it before. As mentioned above about calendars, notes, email, photos, etc. it would actually cause a lot of problems.
In short, no, it's not possible.