I download a file from a website using a Java program and the header looks like below
Content-Disposition attachment;filename=\"Textkürzung.asc\";
The result you're seeing suggests your console expects text to be in Windows "code page 850" encoding - the character ü has Unicode code point U+00FC. The byte value 0xFC renders in Windows code page 850 as ³. So if you want the name to appear correctly on the console then you need to print it using the encoding "Cp850":
PrintWriter consoleOut = new PrintWriter(new OutputStreamWriter(System.out, "Cp850"));
consoleOut.println(filename);
Whether this is what your "other application" expects is a different question - the other app will only see the correct name if it is reading its standard input as Cp850 too.