With SQL server you can just use a CTE instead of IF/THEN logic to make it easy to map from your existing queries and change the number of involved queries;
WITH cte AS (
SELECT product,price,1 a FROM table1 WHERE project=1 UNION ALL
SELECT product,price,2 a FROM table1 WHERE customer=2 UNION ALL
SELECT product,price,3 a FROM table1 WHERE company=3
)
SELECT TOP 1 WITH TIES product,price FROM cte ORDER BY a;
Alternately, you can combine it all into one SELECT
to simplify it for the optimizer;
SELECT TOP 1 WITH TIES product,price FROM table1
WHERE project=1 OR customer=2 OR company=3
ORDER BY CASE WHEN project=1 THEN 1
WHEN customer=2 THEN 2
WHEN company=3 THEN 3 END;