As of July 1st, Japan bans entry of residents from a total 129 countries/regions.

https://www.japan.travel/en/coronavirus/

In the past month, there has been reports the government is considering easing restrictions on business entries from Australia, New Zealand, Thailand and Vietnam.

On July 1st, the Asahi Shimbun reported that Taiwan, South Korea and China are also among the candidates. Brunei and Myanmar could also possibly be included.

After business travel resumes, students will receive the next priority. It will be some time before tourists will be allowed in, the Asahi Shimbun reported.

Following the lifting of the state of emergency, more facilities are starting to reopen to the public.

Some places conduct body temperature checks at the entrance and all locations require visitors to wear a mask.

Places that have reopened:

Amusement Parks

Tokyo Disneyland/DisneySea (reopening July 1st/requires advance reservation)

Gardens

Imperial Palace East Gardens

Rikugien

Koshishikawa Korakuen

Hamarikyu

Shinjuku Gyoen

Obseravtion decks

Tokyo Tower

Tokyo Skytree

Shibuya Sky (Shibuya Scramble Square)

Tokyo City View（Roppongi Hills) 

Tokyo Metropolitan Government Building (from July 1st)

Museums

Tokyo National Museum (requires reservation)

National Museum of Nature and Science（requires reservation)

Miraikan (requires reservation)

Edo-Tokyo Museum

The National Arts Center Tokyo 

Other

Tama Zoo 

Ueno Zoo (requires reservation)

Tokyo Sea Life Park/Kasai Rinkai Aquarium (requires reservation)

Toyosu Market (Early morning tuna auctions remain closed to public)

Digital Art Museum teamLab Borderless (requires reservation)

The much anticipated season for NPB baseball and J League soccer will resume soon. Both leagues announced its revised schedule on the 15th.

・After completing several rounds of preseason games, NPB will begin its regular season on June 19th and play a 120 game season. Games will not go beyond the 10th inning, and each team can carry a max of five foreign players on the roster instead of four.

In the Central League, the playoffs will not be held  and the winner of the league will automatically advance to the Japan Series, scheduled to start on November 21st. For the time being, fans will not be allowed to attend the games.

・The J League 1st division resumes its season on July 4th. For the first week, fans are not allowed in. Beginning on July 10th, up to 5,000 fans will be accepted. The next phase comes on August 1st, when teams will be allowed to sell tickets up to 50% of the stadium capacity.

Teams won't face relegation, while the top two teams from the 2nd division will automatically get promoted if they meet the league requirements.

The Tokyo Tower becomes one of the first tourist facilities to reopen following the lifting of the state of emergency.

And the following rule might allow for the facility to easily implement social distancing rules ; visitors will not be allowed to use the lift and will need to climb up and down the emergency staircase. 

There are about 600 steps to the main observation deck located at the 150 meter level.

Update (June 10)

It has been reported that visitors are already allowed to use the elevators.

There is no discount for visitors that choose to use the stairs.

The state of emergency for the Tokyo metro area and Hokkaido was lifted on May 25, following a drop in new cases of COVID-19.

Many businesses that shut down during the 7 week "soft lockdown" are starting to resume operations with reduced hours and social distancing measures. A major barbershop chain resumed services in most of their shops with staff wearing transparent face shields in addition to masks. 

NPB was quick to announce its pro baseball season will start on June 19th and it hopes to play a 120 game season. The games will initially be played without letting in spectators. 

There is still a voluntary restraint order for non-essential domestic travel, which will be lifted on June 19th. To help the tourism industry, the government announced the "GoTo Campaign" offering coupons that will discount up to 50% of accommodation costs.

Meanwhile it is likely international travel to Japan will take time to resume. Neighboring Asian countries with low infection rates such as Taiwan, Vietnam or Thailand could be candidates of countries that will first be allowed into Japan, but there has been no mention whatsoever of specific dates or necessary guidelines.

Note (May 27): Some foreign media reported the travel subsidy is targeting international visitors to Japan but the Japan Tourism Agency mentioned these reports are inaccurate - the campaign is intended solely for the residents of Japan with the aim of stimulating domestic travel. Residents of total 110 countries are banned from entering the country at the moment.

Effective May 20th, passengers wishing to bring  items with overall dimensions of over 160 cm into the Shinkansen running on the Tokaido-Sanyo route (Nozomi/Hikari/Kodama/Mizuho/Sakura) will need to reserve space prior to boarding the train.

Passengers with oversize luggage who did not make prior arrangements could be charged 1,000 JPY depending on how strict the train conductors decide to enforce this new rule.

The new requirements do not apply to the northbound Shinkansen operated by JR East, such as Hayabusa or Kagayaki.

Japanese long-distance trains have limited space to store large items. During peak travel times, finding sufficient space for luggage has been a consistent problem.

JR has been making improvements to its online booking platform and travelers on wheelchairs can now apply for tickets online.

For Nozomi/Hikari/Kodama trains

https://railway.jr-central.co.jp/pwd/wheelchair/#w4

For JR East operated trains (for Kanazawa, Niigata, Tohoku region and Hokkaido)

https://www.jreast.co.jp/equipment/equipment_1/wheelchair/

Users can submit their request for a specific train and the train company will reply back. Travelers will still need to pick up the physical ticket before travel, since E-tickets are not available for Shinkansen. Also the websites are only in Japanese at the moment.