Just to get out of the box for a moment, have you thought about using WCF with MSMQ? I am not sure that I fully understand you're architecture, but it sounds like the client of the API has to spin up another process that hosts the API. There may be a timing issue between when you start the API host, and when the client tries to make calls. Microsoft has recently deprecated .NET Remoting (as well as all their other previous communications technologies such as ASMX web services) as legacy, and is highly recommending that developers move to the WCF platform.
If you use WCF with MSMQ, you shouldn't have a problem with timing. Your client application can drop messages in a durable queue regardless of whether the API host is running or not. The API host can be started at any time, and it will pick up and process any messages waiting in the queue. Even if you still have the client application start up the API host, the timing problem would no longer be an issue because you are using queuing to transfer the messages rather than .NET Remoting. WCF provides a nice, convenient, easy to use wrapper around MSMQ, so the barrier to entry is relatively low.
The other beautiful thing about using WCF over .NET Remoting is that you can easily move the API host to a different physical server without having to change the client app(s). You could even move to a different queuing platform if you so desired (such as RabbitMQ on AMQP), without changing either the client or API host apps. WCF handles all of that interaction for you, providing a much cleaner decoupling and more dependable communications between your client app and API host.
If moving to WCF is not an option, you should be able to explicitly set the port with .NET Remoting. I am not sure how you are configuring your API host, but the URL for any given remoted object is usually in the form of:
tcp://<hostname>[:<port>]/<object>
If you add the port, then you should be able to use Abhijeet's solution to determine if the port is open or not. You won't gain the loose coupling and dependable communications benefits of WCF, but it would definitely be less work. ;)