How to extract a specific value from a LinkedHashMap to insert it into an array

Just a few general pointers first:

• When you use the word 'hash' in your question you really mean 'map'. Using a hash is a common way to implement a map but it is not the only way (a search tree is another common technique). Best to use the name of the structure rather than the implementation
• Similarly, you should declare your variables using the appropriate interface rather than the implementation class. See below for examples
• records have been introduced to Java. If you are using a recent version of Java they are a good way to define straight data records such as your Info
• there are methods in Map to make it easy to handle missing keys without using if statements

So making those changes:

record FlightInfo(String destination, double departureTime, double arrivalTime) { }

Map<String,List<FlightInfo>> airportFlightMap = new HashMap<>();

public void addFlight(String origin, String destination, double departureTime, double arrivalTime) {
    airportFlightMap
        .computeIfAbsent(origin, o -> new ArrayList<>())
        .add(new FlightInfo(destination, departureTime, arrivalTime));
}

Even better would be to have an Airport class that contains the flights but that's beyond the scope of your question.

when I attempt to extract the departure and arrival time of each flight leaving the airport to insert them into the array I cannot. Is there a way to extract a specific value from a linked list with multiple values in each cell inside of a hash or should I go about it another way?

Taking your question literally, the way to extract the departure and arrival time of each flight leaving an airport you would do:

List<Double> departureTimes = airportFlightMap.get(airportName)
    .stream().map(FlightInfo::departureTime).collect(Collectors.toList());
List<Double> arrivalTimes = airportFlightMap.get(airportName)
    .stream().map(FlightInfo::arrivalTime).collect(Collectors.toList());

Though I'm not sure how that gets you closer to your goal of finding the fastest path from one airport to another. For that you would need a search algorithm. For example, in pseudo code

find path(current route, destination):
    if end of current route = destination:
        process path
    else
        for each flight from end of current route:
            if next airport not in current route:
                find path(current route + next airport, destination)

In that algorithm you probably want to store the origin and a list of FlightInfo as the current route. That way you can easily compare the arrival time to find the best route.