Get to Nijo Castle by 8:00 or 8:30 am. The tour buses start arriving about 9 and the guided groups will start filing through shortly after. If you can stay ahead of them you can see the place in relative peace.

Actually, that is pretty true of most of the major sights. The buses start arriving early to mid-morning. Best to get your most important places done early and just accept the crowding elsewhere later in the days. In July, you will want to go slow or get inside in the afternoon to avoid the heat. You might want to bring sun hats or small, opaque umbrellas to keep the sun off. You can buy very cheap throw-away umbrellas at convenience stores but they are usually clear plastic.

The Imperial Palace is not worth the time in your limited visit plus there are no English language tours on weekends anyway. Best sights for such a compact time might include Kiyomizu-dera, Nijo Castle, Golden Pavilion (Kinkaku-ji), Pottery Slope leading to Kiyomizu-dera (Sannenzaka Street), Heian Jingu Shrine, Ryoan-ji, Nazen-ji. Sanjusangendo Hall would be good in the afternoon as it is always dark and cooler inside. Gion would be nice for an evening stroll. Wear comfortable walking shoes and soft, sturdy socks. Many temples and shrines and the Nijo Castle main building require you to remove shoes before entering.

For only 2 days in July with 3 people, I'd just bite the bullet and move around like many Japanese tourists do, by cab. Spend a bit more to maximize your time and get a little air conditioning between sites.