A char* in C is, as you noted, pointing to the start of your character array (which is how C manages Strings).
In C the size of a char is one byte, and pointers always point to the start of a byte. Your C String is an array of characters, so adding 1 to a pointer moves the start of your string right by one character.
That means that the C code:
char *a;
// Set the String here
a = a + 1;
translates in Java to something like:
String a;
// Set the String here
a = a.substring(1);
or if you are using a char array:
char[] a;
// Set the array contents here
char[] copyTo = new char[a.length];
System.arraycopy(a, 1, copyTo, 0, a.length);
a = copyTo;
Java will be a bit more careful of protecting you that C will be though. For instance, if you have a zero length string, the C code has the potential to either segfault (crashing the application) or give you a gibberish string full of memory junk (then, eventually, crash the application), whereas the Java code will throw an exception (normally an IndexOutOfBoundsException) which you can, hopefully, handle cleanly.
Remember though, that String in Java are immutable. You cannot change them, you can only create new Strings. Fortunately, String has several built in functions which allow you to do a lot of the standard actions, like replace part of the String with another and return the result. A character array is mutable, and you can change the characters within them, but you will lose a lot of the nice benefits you get from using the proper String class.
String?