How can I sign a file using RSA and SHA256 with .NET?

RSA + SHA256 can and will work...

Your later example may not work all the time, it should use the hash algorithm's OID, rather than it's name. As per your first example, this is obtained from a call to CryptoConfig.MapNameToOID(AlgorithmName) where AlgorithmName is what you are providing (i.e. "SHA256").

First you are going to need is the certificate with the private key. I normally read mine from the LocalMachine or CurrentUser store by using a public key file (.cer) to identify the private key, and then enumerate the certificates and match on the hash...

X509Certificate2 publicCert = new X509Certificate2(@"C:\mycertificate.cer");

//Fetch private key from the local machine store
X509Certificate2 privateCert = null;
X509Store store = new X509Store(StoreLocation.LocalMachine);
store.Open(OpenFlags.ReadOnly);
foreach( X509Certificate2 cert in store.Certificates)
{
    if (cert.GetCertHashString() == publicCert.GetCertHashString())
        privateCert = cert;
}

However you get there, once you've obtained a certificate with a private key we need to reconstruct it. This may be required due to the way the certificate creates it's private key, but I'm not really sure why. Anyway, we do this by first exporting the key and then re-importing it using whatever intermediate format you like, the easiest is xml:

//Round-trip the key to XML and back, there might be a better way but this works
RSACryptoServiceProvider key = new RSACryptoServiceProvider();
key.FromXmlString(privateCert.PrivateKey.ToXmlString(true));

Once that is done we can now sign a piece of data as follows:

//Create some data to sign
byte[] data = new byte[1024];

//Sign the data
byte[] sig = key.SignData(data, CryptoConfig.MapNameToOID("SHA256"));

Lastly, the verification can be done directly with the certificate's public key without need for the reconstruction as we did with the private key:

key = (RSACryptoServiceProvider)publicCert.PublicKey.Key;
if (!key.VerifyData(data, CryptoConfig.MapNameToOID("SHA256"), sig))
    throw new CryptographicException();

The use of privateKey.toXMLString(true) or privateKey.exportParameters(true) aren't usable in a secure environment, since they require your private key to be exportable, which is NOT a good practice.

A better solution is to explicitly load the "Enhanced" crypto provider as such:

// Find my openssl-generated cert from the registry
var store = new X509Store(StoreLocation.LocalMachine);
store.Open(OpenFlags.ReadOnly);
var certificates = store.Certificates.Find(X509FindType.FindBySubjectName, "myapp.com", true);
var certificate = certificates[0];
store.Close();
// Note that this will return a Basic crypto provider, with only SHA-1 support
var privKey = (RSACryptoServiceProvider)certificate.PrivateKey;
// Force use of the Enhanced RSA and AES Cryptographic Provider with openssl-generated SHA256 keys
var enhCsp = new RSACryptoServiceProvider().CspKeyContainerInfo;
var cspparams = new CspParameters(enhCsp.ProviderType, enhCsp.ProviderName, privKey.CspKeyContainerInfo.KeyContainerName);
privKey = new RSACryptoServiceProvider(cspparams);