"The tourist has many pockets."
Translation:A turistának sok zsebe van.
You would never make a 3rd person singular possessive by adding -eje.
For a word like zseb that ends in a consonant, the 3rd person singular possessive ending will be either -a /-e or -ja / -je.
There's usually no way to tell just by looking at the word whether it will take the j or not. It's just something you have to memorize as part of learning the word.