I know this is a frequent problem, I myself confronted with this problem many times, yet somehow each Autocad dataset I received is different (although they're from the same area). To some data, the projection I gave them worked perfectly, and in this case it didn't.
So, I've recently received some AutoCad data. Converted them in ArcMap. They are missing spatial reference. But I know their location. They're suppose to be in a specific city in Romania.
So first thing I did, I gave them the Stereo 70 projection (Romania's national projection), I assumed this is the projection they were created in, the data gets somewhere in the middle of the Mediterranean Sea. Second thing I tried is is giving them WGS84, and they got somewhere in the Atlantic Ocean 😉
PS: I use ArcGIS 10.1
PS2: I've tried all 3 conversion methods (I know) in ArcMap.
PS3: The autocad data came with no spatial reference. I only know they should be in a specific city.
Best Answer
As you have mentioned in the comments, your old and new CAD files do not overlap, when imported in ArcGIS. This means, that they do not have a consistent coordinate system.
You could either correct them in AutoCAD, which is a better option, since their topology will be maintained, or you could import them into a geodatabase, and then use Spatial Adjustment to align the old data with the new data.