Climbpro for Garmin Edge 520?

Very, very, very unlikely. The last software update to 520/520+ was almost 2 years ago, the 520+ was replaced by the 530 over 3 years ago and the 530 is likely to be itself replaced in the near future so, as much as I loved my 520 until it stopped working I think you could  start saving up for a 540...which will have climbpro, as does the 530 which will be cheaper as the new one approaches

Thanks for your answer @L.Rouge

And, do you know any app  with the same functions that Climbpro?

I searched in Garmin Connect IQ but nothing...

The standard Elevation Screen is probably as best as you will get on the 520, it is limited by not being able to set the scaling on the X & Y axis but it still shows the climb profile for the section of the course you are on currently plus a few relevant data fields, give it a try. Search on the 520 forum for posts relating to the elevation screen for discussion of its limitations and usage.

There is always the free legs’n’lungs app :

Moderate grade- legs feel it;

Steep grade- legs hurt;

Very steep grade- walking;

Long climb- huffing and puffing. :-)

The 530 is already being discounted and the 130+ also has Climbpro, so it might be a good time to upgrade.