Making the words print in the middle of the Gui.TextField?

If you don't want to create a custom skin you can copy the TextField style and modify it as you need. GUIStyle.alignment is what you need to set to TextAnchor.MiddleCenter.

 //C#
 GUIStyle myStyle = new GUIStyle(GUI.skin.textField);
 myStyle.alignment = TextAnchor.MiddleCenter;
 GUI.TextField(new Rect(TextFieldScreenWidth,TextFieldScreenHeight,TextFieldWidth, TextFieldHeight), "4359465779899875454", 25, myStyle);

By the way: It seems you use a TextField just to display a text. A TextField is meant as input field where the user can input text. To only display text you should use a Label. You still can use the TextField style so it looks like a TextField but it's a label.

That works with all styles.