If you need to keep it sequential and compact in the array, and sized only large enough to hold the items within it, then you'd be better of with a second array. Simply iterate through the array that exists, and push onto the new array. If you hit the matching item to remove, then skip it, and keep going. If the end result is that no item was removed, than return the original, otherwise return the new.
This said, if you keep only an array large enough to hold a given number of items at the time, than its not really optimized. It's quite a bit of effort to copy an array into a larger one, than it is to leave unused space.
Compare it to the add. The remove should be pretty much the exact opposite. If you create an array of current.length + 1, then add an item to it, then it makes sense to do the same with remove. If the add is not always incrementing the current array and creating a new one, than I'd suggest that the remove should not and instead simply setting the offset when found to null.
I think I see what you're getting at here.
Is this for an assignment? If so, you'll want to go over the instructions very carefully. If it specifies that add must append to an array or insert at the end, then yes you'll need to figure out the best way of shuffling back. If it does not specify this, and it does not specify a sort order for the array, than I'd suggest insertion via iteration to find the first null entry. I would perceive this currently as tracking a count of the known values, and adding based on the known number of records.
You can shift back the array as well. Its a matter of determining what you want to remove, and then from that point on reassigning the values back by one (or going the other way and looking forward). You'll need to deal with the null insertion at the end though, otherwise it would replicate the last value in the list. So you can do this within a single loop iteration as well.
If this isn't for an assignment, than you don't deal with arrays at all. You deal with a collection designed for volume insert/remove operations on it. Suggested collection types would be LinkedLists if order doesn't matter (ie in order only), or a Tree if order is important. There is no built in Tree ADT in Java though, so you'd need to write one (or find one online). Linked types are the most ideal since you don't need to do anything special to free up space. Simply remove the item, and insert wherever.