First Japanese Pro Baseball Game?

Yes, I can.

The first professional baseball tournament was held in the Spring of 1936. It was a set of 3 tournaments, the first one held at Koshien between April 29 to May 5, 1936, each of 6 teams playing each other. The second tournament was the Narumi Taikai, held in Nagoya between May 16 to May 17, where each of the remaining five teams (Kingyo dropped out after winning the Koshien Taikai) playing 2-3 games. The third tournament was the Takaratsuka Taikai (held in Hyogo prefecture?), with each team playing two games between May 22 to May 24.

Interestingly, the Giants are listed as members of this league, but did not participate in any of the three tournaments.

Here is a set of links that could be useful.

Source: The Official Baseball Encyclopedia 2004.

The first pro league game* occurred at 11:10 AM on April 29 at Koshien. The two teams were Nagoya (now the Chunichi Dragons) and Dai Tokyo.

The starting batteries (Pitcher-Catcher) were Kiyoshi Makino and Bucky McGaillard (from California) for Nagoya, Hisashi Kondo and Yukinobu Murakawa for Dai Tokyo. Hawaiian Herbert North came on in relief in the first inning for Nagoya and got the win.

McGaillard or Yoshio Takahashi, a Nisei from Hawaii, was first foreign pro league player in Japanese baseball history.

Line Score

Nagoya    0 3 0 0 3 0 2 0 0 - 8
Dai Tokyo 1 0 0 1 0 0 2 0 1 - 5

Adding to what Westbay-san already stated;
1936 Spring Teams
Hankyu > Blue Wave > Orix Buffaloes (present)
Nagoya > Dragons (present)
Kinko went defunct
Dai Tokyo > Whales > BayStars (present)
Senators went defunct
Tigers (present)
Note: the Giants were in America.

1936 Spring Tournaments

Tournament  | Pref. | Dates       | Teams | Style                | Champion
Koshien     | Hyogo | 4/29 - 5/5  |   6   | Round-Robin (1 game) | Senators
Narumi      | Aichi | 5/16 - 5/17 |   5   | Tournament           | Senators
Takaradzuka | Hyogo | 5/22 - 5/24 |   6   | Partial Round-Robin  | Senators, Hankyu

In the Takaradzuka tournament each team played two games versus two other teams with the results totaled.

Kinko (Kingyo?) didn't play in the second tournament. As an aside, this does not make a whole lot of sense as the team was based in Nagoya (the location of the tournament). My best guess is that either the team or parent company (the Nagoya Shimbun, now Chunichi) sponsored the tournament and the team didn't participate to avoid claims of bias or that it was sponsored by a rival paper (possibly Nagoya's parent, the New Aichi Shimbun) and the team or parent refused to participate.

Narumi Tournament

:::::::May 16:::::::\:::::::May 17:::::::
                    \
Senators++++++++++++++++
         3|         \ 5+
G1        |         \  +
         2|         \  +
Tigers----|         \  +
                    \G4++++++++
Hankyu----|         \  |     6+
         1|         \  |      +
G2        |         \  |      +
         7|         \ 2|      +
Nagoya+++++++++++++++++|  G5  +++SENATORS
                    \         |
                    \         |
(Bye) Dai Tokyo+++  \         |
                7+  \        5|
G3               +++++++++++++|
                6|  \
(L2) Hankyu------|  \
                    \  "LOSERS BRACKET"
                    \   (L1) Tigers--|
                    \               3|
                    \         G6     |
                    \               2|
                    \   (L3) Hankyu--|

Following the spring season there were three tournaments in July and six more in the Fall (Sept.-Dec.), with a League Championship series in December

Sources: Tumnet Dragons, The Official Baseball Encyclopedia 2004, The History of Professional Baseball, 1934-2004
* Ignoring the earlier pro clubs from the 1920s.

Hope this helps.
Michael Eng