Reverse String : Recursion
You are not using any stored data in the RevereseString [sic] object, so you do not actually need to create the object.  Changing reverseStr() to a static method, (optionally private as well) and you can remove RevereseString reverseStr = new RevereseString(); to save one line of code.

There is no need for a midIndex.  You just need to reverse characters until the startIndex and lastIndex meet or cross.  Thus you can remove int midIndex = 0 + (oldArray.length - 0)/2 ; and save another line of code.

Finally, you are not returning a new array; you are returning the original oldArray and assigning that object to newArray.  In other words, as main() finishes, oldArray == newArray is true.  You have reversed the oldArray in place; there is no need for a return value from reverseStr(), so you can eliminate at least another two lines of code.