A bit simpler version:

select  *
,   ISNULL(max([Payment in installments]) over(partition by customernumber order by date ROWS BETWEEN UNBOUNDED PRECEDING AND 1 PRECEDING), 0)
from (
    VALUES  (1, 2, 0, N'2022-01-02')
    ,   (2, 2, 0, N'2022-02-10')
    ,   (3, 2, 1, N'2022-04-05')
    ,   (4, 3, 0, N'2022-06-01')
    ,   (5, 2, 0, N'2022-06-08')
    ,   (6, 2, 0, N'2022-08-22')
    ,   (7, 2, 1, N'2022-10-03')
    ,   (8, 3, 0, N'2022-11-04')
    ,   (9, 2, 0, N'2023-01-04')
) t (Id,CustomerNumber,[Payment in installments],Date)
order by id

By using ROWS BETWEEN UNBOUNDED PRECEDING AND 1 PRECEDING you can skip the current row's value and get previous ones

This can be done using lag() window function to get the previous row, and sum() window function to retrieve a cumulative sum based on the previous_installement value :

with cte as (
  select *, lag(Payment_installments, 1, 0) over (partition by CustomerNumber order by Date) as previous_installement
  from mytable
),
cte2 as (
  select *, sum(previous_installement) over (partition by CustomerNumber order by Id) as sm
  from cte
)
select Id, CustomerNumber, Payment_installments, Date, case when sm > 0 then 1 else 0 end as HasInstallmentPaymentInThePast
from cte2
order by Id

Demo here

You can try using a subquery and Window function for solve your problem

first you must find previous_Payment_in_installments with lag in Window function

then you can find HasInstallmentPaymentInThePast with max

select 
         id
        ,CustomerNumber
        ,Payment_in_installments
        ,Date
        ,max(previous_Payment_in_installments) 
                     over (partition by CustomerNumber order by Date) as HasInstallmentPaymentInThePast
from  (
          select 
                  id
                , CustomerNumber
                , Payment_in_installments
                ,Date
                ,lag(Payment_in_installments,1,0) 
                               over (partition by CustomerNumber order by Date) as previous_Payment_in_installments
          from CustomerPurchaseInfo
)s 
order by id

You can to insert the basic data with the following codes

drop table if exists CustomerPurchaseInfo

create table CustomerPurchaseInfo (Id int,CustomerNumber int,Payment_in_installments int,Date date  )
insert into CustomerPurchaseInfo
(id,CustomerNumber,Payment_in_installments,Date)
           select 1 as Id ,2    as CustomerNumber,0 as Payment_in_installments,'2022-01-02' as Date
union all select 2  as Id ,2    as CustomerNumber,0 as Payment_in_installments,'2022-02-10' as Date
union all select 3  as Id ,2    as CustomerNumber,1 as Payment_in_installments,'2022-04-05' as Date
union all select 4  as Id ,3    as CustomerNumber,0 as Payment_in_installments,'2022-06-01' as Date
union all select 5  as Id ,2    as CustomerNumber,0 as Payment_in_installments,'2022-06-08' as Date
union all select 6  as Id ,2    as CustomerNumber,0 as Payment_in_installments,'2022-08-22' as Date
union all select 7  as Id ,2    as CustomerNumber,1 as Payment_in_installments,'2022-10-03' as Date
union all select 8  as Id ,3    as CustomerNumber,1 as Payment_in_installments,'2022-11-04' as Date
union all select 9  as Id ,2    as CustomerNumber,0 as Payment_in_installments,'2023-01-04' as Date