Effect of changing data types with row compression enabled
As the documentation you alredy have linked states, ROW compression uses only the bytes that are needed. Once ROW compression is in use, the CPU cycles used to convert from/to int or bigint are the same: I would not worry about that.

BTW, if you're not sure whether int/bigint has an impact on the database size or not (it hasn't), you can see for yourself with a quick and dirty repro:

USE tempdb;
GO

CREATE TABLE SomeTable (
    SomeColumn bigint
)
GO

ALTER TABLE [dbo].[SomeTable] REBUILD PARTITION = ALL WITH (DATA_COMPRESSION = NONE);

INSERT INTO SomeTable 
SELECT TOP 10000000 ROW_NUMBER() OVER (ORDER BY(SELECT NULL))
FROM sys.all_columns AS A1
CROSS JOIN sys.all_columns AS A2;
GO

-- Rebuild the heap, so that pages compact nicely
ALTER TABLE dbo.SomeTable 
REBUILD 
WITH 
(
    MAXDOP = 1, 
    ONLINE = OFF,
    FILLFACTOR = 100,
    PAD_INDEX = OFF
);
GO

SELECT SUM(page_count)
FROM sys.dm_db_index_physical_stats(
        DB_ID(),
        OBJECT_ID('SomeTable'),
        DEFAULT,
        DEFAULT,
        'detailed'
    ) AS ips;


-- 21009 pages used


ALTER TABLE [dbo].[SomeTable] REBUILD PARTITION = ALL WITH (DATA_COMPRESSION = ROW);
GO


SELECT SUM(page_count)
FROM sys.dm_db_index_physical_stats(
        DB_ID(),
        OBJECT_ID('SomeTable'),
        DEFAULT,
        DEFAULT,
        'detailed'
    ) AS ips;

-- 13587 pages used


ALTER TABLE SomeTable ALTER COLUMN SomeColumn int;

-- Rebuild the heap, so that pages compact nicely
ALTER TABLE dbo.SomeTable 
REBUILD 
WITH 
(
    MAXDOP = 1, 
    ONLINE = OFF,
    FILLFACTOR = 100,
    PAD_INDEX = OFF
);
GO


SELECT SUM(page_count)
FROM sys.dm_db_index_physical_stats(
        DB_ID(),
        OBJECT_ID('SomeTable'),
        DEFAULT,
        DEFAULT,
        'detailed'
    ) AS ips;

-- 13587 pages used (same as bigint)