Short answer: some people may use it idiomatically, but no, you probably shouldn't say "I crossed a tunnel", however not for the reason you suggest.
The compass direction in which you are travelling doesn't really have any bearing on whether or not you should use "cross" or "across". For example, a journey from the UK to the USA is "across the Atlantic", and is broadly an east-to-west journey. However, the journey from the UK to France across the English channel is broadly in a southerly direction, but is still referred to as a "crossing".
Both "cross" and "across" imply travelling from one side of something to the other. But what is on the "side" is all a matter of perspective. Obviously, if you are facing somebody, your left is their right; but what if someone is at 90° to you? Their left is either behind or in front of you. And most people make journeys by going forward! You can safely use "cross" or "across" for a journey in any direction.
It is true that compass directions have some bearing on how we describe journeys across land. For example you might speak of "travelling up" to a place which is north of your starting point, or "down" to a place further south.
But the reason you probably shouldn't say "I travelled across a tunnel" is that the word is synonymous with "over", and one does not travel "over a tunnel" as you do a bridge - you go through it or along it.
It might be more accurate to say you went across whatever it is the tunnel spans. For example:
The channel can be crossed in 5 minutes via the tunnel.
But if you want to keep the tunnel as the subject, a better word might be "traversed", which can mean to travel across or through.
The tunnel can be traversed in 5 minutes.