Select if number range contains number in PostgreSQL

Use a range.

SELECT string_to_array(tuesday, '-')
FROM coverage
WHERE 900 <@ int4range(split_part(tuesday, '-', 1)::int4, split_part(tuesday, '-', 2)::int4, '[]');

That last parameter [] signifies an inclusive range where '100-900' would match. You could also do an exclusive upper range like [) (note the right paren) where '100-900' would not match because the upper number is excluded from the set of matching numbers.

For better query speed as your table gets larger, you can add a GIST functional index.

CREATE INDEX tuesday_range_idx ON coverage
  USING GIST (int4range(split_part(tuesday, '-', 1)::int4, split_part(tuesday, '-', 2)::int4, '[]'));

This is exposing some weaknesses in your data model. By having each day as a column, you'd have to create a separate functional index for each column. You're also having to parse text into an array every time you run this. Typically you'd want the data in the table to match how you access it, not its serialized form.

Instead of

CREATE TABLE coverage (
  id serial PRIMARY KEY,
  year smallint,     -- tracking by week
  week_num smallint, -- for example
  sunday varchar,
  monday varchar,
  tuesday varchar,
  wednesday varchar,
  thursday varchar,
  friday varchar,
  saturday varchar
);

why not something like

CREATE TABLE coverage (
  id serial PRIMARY KEY,
  day date NOT NULL UNIQUE,
  daily_data int4range NOT NULL
);

INSERT INTO coverage (day, daily_data)
VALUES ('2020-06-02', '[540,1020]');

Then your search looks like

SELECT daily_data
FROM coverage
WHERE extract(DOW FROM day) = 2 -- Tuesday (Sunday is 0, Saturday is 6)
  AND 900 <@ daily_data;

You can make indexes for the daily data ranges, by date (already a unique index in my example), functional indexes for the day of the week, month, year, etc. Much more flexible.

Or if you absolutely want an array back from your SELECT

SELECT ARRAY[lower(daily_data), upper(daily_data)]
FROM coverage
WHERE extract(DOW FROM day) = 2 -- Tuesday (Sunday is 0, Saturday is 6)
  AND 900 <@ daily_data;