I was surprised that he can fulfill his obligation.
What confused me is the clause, being nominal, unable to modify neither "surprised" nor main clause in this sentence
Let's look at the active voice equivalent of this sentence:
That he can fulfill his obligation surprised me.
In this construction, the nominative clause is the subject of the matrix clause. The verb is transitive. The pronoun is the direct object of the transitive verb. Let's substitute a pronoun for the nominative clause:
Something surprised me.
When we transform this to the passive voice, it looks a little different than your original:
I was surprised by something.
The "by something" is an adverbial prepositional phrase which modifies the passive voice verb. Let's also look at one more sentence:
I was surprised because he can fulfill his obligation.
In this case we have an explicit adverbial clause.
The subordinate clause in your original sentence is nominative in structure, but it's certainly adverbial in use. Is this a strange exception, or are there other nominative constructions that serve adverbial purposes? How about a simple noun?
I was surprised today.
"Today" is a noun. It happens to be a noun that can also serve an adverbial purpose. "That he can fulfill his obligations" is a nominative clause. It happens to be a clause that can also serve an adverbial purpose. Although we can't use every nominative construction in an adverbial capacity, we also can't assume that having a nominative structure prevents an adverbial use.