It is not possible to use the indicative I'm afraid. With this construction: "No creer que ..." we always need a subjunctive. In this specific sentence it actually has a slightly different meaning to the general translation "I don't think that...". It means "I doubt that..." which triggers the subjunctive. "Dudo que vaya al cine después" = "No creo que vaya después al cine"
About having the same subject in both clauses: "(yo) no creo que (yo) vaya al cine después", the rule doesn't apply here in this cases. You can apply that rule to verbs that express desire or hope, for example: "Quiero ir al cine" vs "Quiero que vayas al cine" or "Espero llegar a tiempo" vs "Espero que llegues a tiempo". Notice thought that in the cases with the same subject you don't have "que" but it is followed by an infinitive.
I hope this clarified your doubt.
Gracias por tu pregunta