How to use Rank Function With Condition in Postgres

For the following sample table I want to set ranks to each group order by start_date desc but I want the admin to be always set to rank 0

Expected result:

SELECT name, start_date, group_no, 
RANK() OVER (PARTITION BY group_no ORDER by start_date DESC) as rnk
FROM Table_Name
ORDER BY rnk;

The above query gives me the following result. Is there a way I can add a condition on rank function to set admin to 0 by default and apply rank function to everything else.

I have also tried using case condition:

SELECT name, start_date, group_no, 
CASE WHEN name = 'Admin' THEN 0 
ELSE RANK() OVER (PARTITION BY group_no ORDER by start_date DESC) 
END as rnk
FROM Table_Name
ORDER BY rnk;

In this instance, Admin gets set to 0 but for the subsequent rows the ranking starts from 2 instead of 1.

The other option that I have is to use the following which does the job but is there a better approach to this?

SELECT name, start_date, group_no, 
0 as rnk
FROM Table_Name
WHERE name = 'Admin'
UNION 
SELECT name, start_date, group_no, 
RANK() OVER (PARTITION BY group_no ORDER by start_date DESC) as rnk
FROM Table_Name
WHERE name <> 'Admin'

The following query produces the OP's expected results:

SELECT
  name,
  start_date,
  group_no,
  DENSE_RANK()
    OVER (PARTITION BY group_no
          ORDER BY CASE WHEN name <> 'Admin' THEN start_date END DESC)
  - BOOL_OR(name = 'Admin') OVER (PARTITION BY group_no)::INTEGER AS rnk
FROM
  table_name;

DENSE_RANK is used instead of RANK so that multiple occurrences of name = 'Admin' or of start_date don't introduce gaps in the rankings. The result is that rnk represents recency within a group with rows where name = 'Admin' always considered the most recent regardless of date. Note that this query will rank NULL start_date as more recent than valued start_date.

The following version removes the second window function (as suggested by @ysth):

SELECT
  name,
  start_date,
  group_no,
  CASE WHEN name = 'Admin' THEN 0
       ELSE DENSE_RANK()
              OVER (PARTITION BY group_no
              ORDER BY name IS NOT DISTINCT FROM 'Admin', start_date DESC NULLS LAST) END AS rnk
FROM
  table_name
ORDER BY group_no, rnk, name;

Note that this version ranks NULL start_date as less recent than valued start_date.