Reconstruct uint32_t from char array to RTC.set
There are several possible issues here:


can you send the number, as a string, from the PC to the ATtiny?
can you convert the string to an actual number?
can you give this number to the RTC?


The phrasing of your question sounds like your issue is with the
conversion. However that is easily handled with atol(). Here is a test
program I tried on my Uno:

void setup() {
    char time_string[] = "1532341020";
    uint32_t time_number = atol(time_string);

    Serial.begin(9600);
    Serial.print("string: \"");
    Serial.print(time_string);
    Serial.print("\", number = ");
    Serial.println(time_number);
}


The output is:

string: "1532341020", number = 1532341020