First, I don't understand when you say "they are the same word". You mean new and old are expressed in Japanese as same phonetocally?　
As far as I know, new is "shin　(新）" or "atarashii(新しい）", and old is "kyuu　(旧）" or "furui　(古い）".
Now let's talk about "old". You have noticed I used two characters 旧 and 古, and used them as same.
When you say new and old are same word in Japanese, then you are misinformed. But the informer could have meant that these two things are put in one word.
If that is the case, then my answer 新旧 could be what you are looking for.
This is one word yet have two opposite ideas. 新旧is being used like this:
新旧を比べる = Comparing old and new.

Hope this helps.