How can I instantly update a table in a database with the columns values from another database?
You can probably use a trigger to accomplish your goal.

Consider the following simplified example:

CREATE TABLE t1
(
    c1 int
    , c2 int
);

CREATE TABLE t2
(
    c1 int
    , c2 int
);

CREATE TRIGGER t1_trigger AFTER INSERT ON t1
FOR EACH ROW
BEGIN
    INSERT INTO t2 (c1, c2)
    VALUES (New.c1, New.c2);
END;

INSERT INTO t1 (c1, c2)
VALUES(1, 2);


Check the results:

SELECT *
FROM t2;


╔═════╦════╗
║ c1  ║ c2 ║
╠═════╬════╣
║  1  ║  2 ║
╚═════╩════╝


I've created a simple fiddle that shows the working code, and allows you to play with it. 

The above creates two tables, the first with a trigger that copies rows inserted into t1 into t2.  In the above example, both tables are in the same database... since you are attempting to copy from one database to another, the following syntax might work in the trigger if both databases are on the same MySQL Server:

CREATE TRIGGER t1_trigger AFTER INSERT ON t1
FOR EACH ROW
BEGIN
    INSERT INTO other_database.t2 (c1, c2)
    VALUES (New.c1, New.c2);
END;