Вроде и с пробелами все норм.
package com.javarush.task.task08.task0823;
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
/*
Омовение Рамы
*/
public class Solution {
public static void main(String[] args) throws IOException {
BufferedReader reader = new BufferedReader(new InputStreamReader(System.in));
System.out.println("Введи: ");
String s = reader.readLine();
StringBuilder s1 = new StringBuilder(s);
s1.setCharAt(0, Character.toUpperCase(s1.charAt(0)));
for (int i = 0; i<s1.length(); i++){
if (Character.isSpaceChar(s.charAt(i))){
s1.setCharAt(i+1, Character.toUpperCase(s1.charAt(i+1)));
}
}
System.out.println(s1);
}
}