I hope this will help a little. I dont remember exactly when but i know some time in day,
and it is in salt water.

Also is the pufferfish the big blue one on the shore?

The puffer fish is in the salt water it is really ugly its a brown fist. You can see its
mouth from top. I hope that helps

You can find the pufferfish in the water in front of your sim's house. I forget what time
of day, but I think it is full sun.

The pufferfish you can catch on the beach at your shelter or right from it where the box
appears once a day.
You have to fish when the sun is at his highest point in the sky.
If you look at the clock with the four circles , you have to fish when the arrow is on the
circle which is full yellow.
It doesnt mean you catch a pufferfish the first time,not the seccond,maybe if you are
lucky.
But you have to try many times and then one time they appear and they are not so difficult
to catch if they appear.
I hope you can understand my english,i am speaking dutch so....
ITS A CRUEL GAME , DONT QUIT SO EASY , YOU WILL SUCCEED.