How to write query with cast and switch functions in SQL Server?

Personally, I'd go with a mapping table, but another option is CHOOSE() or even the traditional CASE

Note the +1 in the CHOOSE option ... 0 is not a valid option and would return NULL

Example

Declare @YourTable Table ([Status] int)
Insert Into @YourTable Values 
 (0),(1),(2),(3)

 
Select * 
      ,ViaCHOOSE = choose([Status]+1,'New','Identified','Some Other','...')
      ,ViaCASE   = case [Status] 
                        when 0 then 'New'
                        when 1 then 'Identified'
                        when 2 then 'Some Other'
                        else null  -- or 'Undefined'
                    end
From @YourTable

Results

Status  ViaCHOOSE   ViaCASE
0       New         New
1       Identified  Identified
2       Some Other  Some Other
3       ...         ...

You could create a (temporary) table with that mapping.

create table XYMapping (number int, text varchar(max));
INSERT INTO XYMapping (number, text) VALUES (1, 'New'), (2, 'Identified'); -- ...

Insert all values and then join them.

Steps to follow to convert int column to text and replace the existing values.

1. Alter the table. Since INT converts to text implicitly NOT NULL is able to be maintained
2. Exec UPDATE statement using conversion table specified using VALUES table value constructor

Something like this

drop table if exists #samples;
go
create table #samples (
  id           int not null,
  stat         varchar(10) not null);

insert #samples(id, stat) values 
(10, 'Start'),
(1, 'Start'),
(1, 'Failed'),
(2, 'Start'),
(3, 'Start'),
(3, 'Failed'),
(4, 'Start'),
(4, 'Failed'),
(4, 'Start'),
(4, 'Failed');

/* step 1: alter the table (and implicitly convert) */
alter table #samples
alter column
  id        varchar(20) not null;

/* step 2: update based on conversion table */
update s
    set id=v.new_str
from #samples s
     join 
     (values ('1', 'New'),
             ('2', 'Identified'),
             ('3', 'Identified'),
             ('4', 'Identified'),
             ('10', 'Other')) v(old_int, new_str) on s.id=v.old_int;

select * from #samples;

id          stat
Other       Start
New         Start
New         Failed
Identified  Start
Identified  Start
Identified  Failed
Identified  Start
Identified  Failed
Identified  Start
Identified  Failed