I highly recommend keeping a light first day, especially after flying around the world. If you do have the energy and if you don't take too long getting through Customs & Immigration, you could have your bags sent to your hotel by a luggage forwarder (takuhaibin) and go straight from the airport to sightseeing. Ask them if they can have the bags delivered to your hotel by that evening or not. Costs are not bad, and you already mentioned your time is limited.
For Day 2, Kappabashi is an often overlooked great place - be sure to check out some of the "sanpuru" stores where you can see some incredibly detailed models of food used at restaurant window displays. Some can make some excellent (though pricey) souvenirs. Kappabashi is only a 10 minute walk or so from Sensoji. If you want a good photo spot of Sensoji, go to the tourist info office across the street and then up to the 8th floor.
For Day 3, don't miss the shrine garden in the middle of Meiji Shrine - most people miss it completely, unfortunately. You timing is also good to see the irises in bloom, which are especially nice at the Meiji garden. You also might consider going to Shibuya Sky around sunset - it is not free like the TMG in Shinjuku, but does give a killer view.
As others have mentioned, the East Imperial Garden is OK, but there are far better ones in Tokyo. Rikugien, Kishikawa Korakuen, and Kiyosumi to name a few of them. All are open on Mondays, and they are all beautiful, inexpensive, and normally uncrowded. Horikiri also has a beautiful iris garden.
https://goo.gl/tAECfMI also suggest you check if your itinerary justifies a JR Rail Pass. It may be your agency pushing it so they can get a commission from it. Even if you do break even, the pass also limits you to non-Nozomi trains (Nozomi trains are far more numerous and convenient) unless you pay an additional surcharge.
Be aware that the rainy season starts in June. The first part of the month is not that bad at all, but from around mid-June you could get hit with days of heavy rain. It also gets very humid and it's like walking in a steam bath all day.