What are the main obstacles of building bridges using carbon (nano) tubes? When could we overcome those obstacles? Would infrastructure built using this technology together with concrete be cheaper and more robust than steel and concrete bridges?

I’m not sure these are the ”main” obstacles, but certainly major ones.

Supply.

A bridge or any infrastructure project would require many many many times more carbon fiber than other objects that use it. A single bridge would likely require the entire annual output or more just for the bridge due to volume alone.

The filler/matrix.

Carbon nanotubes are so long compared to their cross-section that they are considered to be a tension only material. To solve this they are placed/woven into resin, or plastic, or some other type of material. But, none of these materials come close to the strength of the carbon fiber especially when compared to the weight of the resin. So, in high strength applications you’re still looking at tension only applications.

Cost.

At the sheer scale of material involved you can’t even come close to how cheap steel and concrete are compared to carbon nanotubes. By volume, the cost is thousands of times more expensive to build with carbon nanotubes.

There are a number of other items I have heard brought up but I have less knowledge about:

Abrasion resistance – Either the nanotubes themselves or the matrix that holds them.

Fire resistance – I believe the concern is the matrix material degrading and no longer holding the nanotubes.

We seem like we have a long way to go on this material before it becomes a primary construction material.