Terraria – How to transfer a Terraria character/world without using the original laptop

Since the only way to run Terraria on Ubuntu is through WINE as Ben Brocka and lunboks have suggested, I will answer your actual question, assuming that you are using WINE and if you are not, I would suggest that you revise your question to provide more information about your setup.

While I'm sure you knew this if you backed up your files, but to provide background for the answer, as per this question, the default behaviour is for saves to be in My Documents\My Games\Terraria\Players and worlds to be in My Documents\My Games\Terraria\Worlds as stated here.

You will want to put your character and world files in the paths mapped for My Documents\My Games\Terraria\Players and My Documents\My Games\Terraria\Worlds respectively. The default WINE behaviour if I read correctly is for My Documents to map to your home directory so it would be home/(user)/My Games/Terraria/Players and home/(user)/My Games/Terraria/Worlds for your respective username I believe. If you want and this is up to you, you can remap where My Documents maps to in wineconfig, but you will have to adjust where you put the save files accordingly.

I don't believe there is. Terraria is solely run from Steam, and the only possible way you could run it outside of Steam was if it was Pirated (which is illegal.) You can still play it if you're not connected to the internet, however. You simply run Steam in offline mode.