Grand Theft Auto V – How to Save Progress

It'll be because the saved game is strictly for your profile (your gamertag), you won't need the internet to save, but you first need to download your profile to continue the saved game.

Having your cloud enabled can also cause this problem - try disabling it in Storage settings.

Yes it is possible to play on PS4 online and import your PS3 profile later.

You will be given the option to transfer your previous GTA Online character and progression data, along with the opportunity to change your character’s appearance again. If you wish to complete the transfer later, you can return to this option via the Online section of the Pause Menu.

What will be transferred:

• Money – Earned in-game money will transfer. Purchased in-game money that has not yet been spent will only transfer within a console family (e.g. PlayStation®3 to PlayStation®4).
• Rank and RP – You will have the same Rank and RP after transfer.
• Game Progress – GTA Online progression will transfer including all unlocks, Contacts, Jobs, Trophies and stats. Items and Property – Weapons/attachments, vehicles/modifications, properties, clothing and all aspects of character appearance will be transferred. Character-owned items (e.g. clothing, weapons, etc.) will remain in the character’s inventory and will not need to be repurchased in-game. Vehicles will appear in the same garage they were left in before the transfer.
• Creator – Jobs made within the Creator will be transferred.
• Crews – Existing Crew information will available.
• Achievements and Trophies – Achievements and trophies will transfer. Additional overall Gamerscore will not be awarded for transferring achievement progress.

Keep in mind that any existing character data and progress will be overwritten and that only money earned will be combined with the imported characters. Also character transfer can only be done once per account.

What you need to do and what else you need to know is described on the Rockstar support page.