How to update column coming from TOP 1 of another table

UPDATE City
SET Period = a.AssetID
FROM (SELECT TOP 1 AssetID, Name FROM Assets ORDER BY AssetID ASC) AS a
WHERE City.Name = a.Name;

This should work:

update City
set Period = (
    select top 1 a.AssetID
    from Assets a
    where City.Name = a.Name
    order by a.AssetId asc)

Sample code to test:

create table #City (CityId varchar(20), [Name] varchar(20), Period varchar(20))
create table #Assets (AssetId varchar(20), [Name] varchar(20))

insert into #City values (1, 'Asset1', null)
insert into #City values (2, 'Asset2', null)
insert into #City values (3, 'Asset3', null)

insert into #Assets values (1, 'Asset1')
insert into #Assets values (2, 'Asset1')
insert into #Assets values (3, 'Asset1')
insert into #Assets values (4, 'Asset2')
insert into #Assets values (5, 'Asset2')
insert into #Assets values (6, 'Asset3')
insert into #Assets values (7, 'Asset3')

select * from #City
select * from #Assets

update #City
set Period = (
    select top 1 a.AssetID
    from #Assets a
    where #City.Name = a.Name
    order by a.AssetId asc)

select * from #City

drop table #City
drop table #Assets