There are a number of helpful, some very detailed, works on Barth and Roman Catholicism. Barth himself had an ambivalent but intensely engaged dialogue with Catholicism throughout his academic career, especially leading up to, during, and after Vatican II. On the one hand he rejected a number of fundamental Catholic doctrines (e.g. the understandings of grace, salvation, church authority vs scriptural authority, Mary, and natural theology - in particular the analogia entis - 'analogy of being'). On the other he regards Catholicism as no worse, and in many ways better, than 19th century Liberal Protestant theology, and was somewhat hopeful for reform, especially at Vatican II. A hope partly rewarded, and not without his own influence (as you seem to have read.)
If you are happy to do some solid reading, one place to start is with the treatment of Barth's theology by his younger Catholic compatriot and friend,
- Hans urs Von Balthasar: The Theology of Karl Barth (the 2nd edition is better than first, being written in 1961 after more of Barth's Church Dogmatics was published.)
Von Balthasar is regarded as something of a vexed interpreter of Barth by many today, but Barth himself spoke highly of him. Here you will see at play something of Barth's influence on one Catholic scholar. Hans Kung also has a work on Barth.
For a more analytical assessment and Barth's early engagement with Catholicism, you may prefer to start with:
- Amy Marga, Karl Barth's Dialogue with Catholicism in Goettingen and Muenster (2010)
- Donald Norwood (ed.), Reforming Rome: Karl Barth and Vatican II (2015), for his later engagement.
There are various summary articles in academic journals also, and a number of works in German, but I'll leave those for you to hunt down, depending on whether you can access and read them.
One side note, most recent scholarship would no longer regard Barth as 'Neo-orthodox', a term regarded as misapplied to Barth. (See for example the works of Bruce McCormack: Karl Barth's Critically Realistic Dialectical Theology and more recent: Orthodox and Modern) Both are relatively dense reading in places, the former particularly, but very instructive all the same.
Of course, far and away the best way to understand Barth's engagement with Roman Catholicism is to read the Church Dogmatics. Here you will both see it in action and get the proper context to his arguments, as well as come to understand Barth much more deeply on a wider basis. It can be a daunting task. But its rewards are manifold and significant.