Before you start, back up the phone using iCloud and/or via iTunes on your Mac.
iOS: Back up and restore your iOS device with iCloud or iTunes
Then set up iCloud on your computer. Your contacts will start to sync to your contact app on the Mac as soon your iCloud account is active and you selected to sync from iCloud in Contacts/Settings/Accounts.
Make sure to have at least one contact info listed in Contacts, if you did not use the app before. Syncing to an empty contact list can erase the contact info instead of adding new info.
After that, activate syncing contacts on the phone in Settings/iCloud
From now on, every time you start the contact app on your Mac or iPhone, iCloud will make sure that the info stays the same on each device. If you edit the contact info on your Mac, the info will be corrected on your phone as soon as you start contacts on the phone.
Good Luck