Transaction in a stored procedure
You need to wrap that code in CREATE PROCEDURE ... syntax, and remove the GO statements after BEGIN TRANSACTION and before COMMIT TRANSACTION.

GO
CREATE PROCEDURE dbo.AssignUserToTicket
(
     @updateAuthor varchar(100)
    , @assignedUser varchar(100)
    , @ticketID bigint
)
AS
BEGIN
    BEGIN TRANSACTION;
    SAVE TRANSACTION MySavePoint;
    SET @updateAuthor = 'user1';
    SET @assignedUser = 'user2';
    SET @ticketID = 123456;

    BEGIN TRY
        UPDATE dbo.tblTicket 
        SET ticketAssignedUserSamAccountName = @assignedUser 
        WHERE (ticketID = @ticketID);

        INSERT INTO [dbo].[tblTicketUpdate]
            (
            [ticketID]
            ,[updateDetail]
            ,[updateDateTime]
            ,[userSamAccountName]
            ,[activity]
            )
        VALUES (
            @ticketID
            , 'Assigned ticket to ' + @assignedUser
            , GetDate()
            , @updateAuthor
            , 'Assign'
            );
        COMMIT TRANSACTION 
    END TRY
    BEGIN CATCH
        IF @@TRANCOUNT &gt; 0
        BEGIN
            ROLLBACK TRANSACTION MySavePoint; -- rollback to MySavePoint
        END
    END CATCH
END;
GO


Also note, I have added a TRY...CATCH statement block to allow performing a ROLLBACK TRANSACTION statement in case some error occurs.  You probably need better error handling than that, but without knowledge of your requirements, that is difficult at best.

Some good reading:


Always specify the schema
Stored Procedure Best Practices
Bad habits to avoid