Change database and execute update query (T- Transact)
One solution would be to use SQL Servers sp_executeslql function. Utilising this function you could write some sql like the sample below. 

DECLARE @DatabaseToUse  VARCHAR(100),
        @SQLToRun       VARCHAR(200);

SELECT  @DateBaseToUse = 'MyDataBaseName';

SELECT @SQLToRun = 'UPDATE ' + @DatabaseToUse  + '.dbo.Table SET Column1 = Column1 + 1';

Execute sp_executesql @SQLToRun;


You should replace the .dbo. with the name of your Schema.