Issue 1:

When i am trying to execute the Batch class.I am getting following error

{"message":"This session is not valid for use with the REST API","errorCode":"INVALID_SESSION_ID"}]

I suspect what you've run into is covered by the idea Batch Jobs - Multi User - Named Credential Usage. There is also the corresponding SFSE question Using Named Credentials in Batch Job (multi user). Basically, the Named Credential isn't working in a batch context if they are per user.

It appears that a Named Credential doing oAuth with a refresh token may be more successful. See Using Named Credentials to get Salesforce sessionId.

Issue 2:

For GET Method is working,only for POST Method is not working?

When I first checked this I didn't have "Advanced Currency Management" enabled in the org. As such, the metadata I could see for DatedConversionRate via /services/data/v43.0/sobjects/DatedConversionRate indicated is neither createable (the typical reason to do a POST call) nor updateable (which would be the PATCH verb).

After enabling "Advanced Currency Management" the DatedConversionRate records became createable and updateable.

Test POST from Workbench to create a new DatedConversionRate for a known currency ISOCode on a specific StartDate:

POST /services/data/v43.0/sobjects/DatedConversionRate

{
  "IsoCode" : "GBP",
  "ConversionRate" :  0.667500,
  "StartDate" : "2018-07-16"
}

201 Response

{
  "id" : "04w1W000000k9dOQAQ",
  "success" : true,
  "errors" : [ ]
}

Next steps in debugging...:

1. Do the same POST from Anonymous Apex with the local Session ID
2. Do the same POST from within a batch with an established Session ID (login API)
3. Do the same POST from Anonymous Apex with a Named Credential
4. Do the same POST from within a batch with a Named Credential

Do the same POST from Anonymous Apex with the local Session ID

public class DateConversionRateCreate {
    public void create(string isoCode, decimal conversionRate, DateTime startDate) {
        
        map<string, object> mapToSerialize = new map<string, object>();
        mapToSerialize.put('IsoCode', isoCode);
        mapToSerialize.put('ConversionRate', conversionRate);
        mapToSerialize.put('StartDate', startDate.format('yyyy-MM-dd'));
        string jsonstring = JSON.serialize(mapToSerialize);
        
        Http h = new Http();
        HttpRequest req = new HttpRequest();
        // Use the "local" session ID from the active transaction
        req.setHeader('Authorization', 'OAuth ' + UserInfo.getSessionId());
        req.setHeader('Content-Type', 'application/json');
        req.setBody(jsonstring);
        // Hardcoded endpoint for expediency. Has been added as a remote site.
        req.setEndpoint('https://na5.salesforce.com/services/data/v43.0/sobjects/DatedConversionRate');
        
        
        req.setMethod('POST');
        System.debug(LoggingLevel.INFO,'request body:'+req.getBody());
        
        HttpResponse res = h.send(req);
        System.debug(LoggingLevel.INFO,'res'+res.getBody());
        System.debug(LoggingLevel.INFO,'res'+res.getStatus());
    }
}

Run with anonymous Apex:

DateConversionRateCreate dcrc = new DateConversionRateCreate();
dcrc.create('GBP', 0.667400, DateTime.now());

Response

02:06:26.1 (19329463)|USER_DEBUG|[19]|INFO|request body:{"StartDate":"2018-07-18","ConversionRate":0.667400,"IsoCode":"GBP"}
02:06:26.1 (637348591)|USER_DEBUG|[22]|INFO|res{"id":"04w1W000000k9dTQAQ","success":true,"errors":[]}
02:06:26.1 (637464103)|USER_DEBUG|[23]|INFO|resCreated

Do the same POST from within a batch with an established Session ID (login API)

I used a variation of the above code, but passed the Session ID into the constructor. This works for a quick test, but isn't recommended for actual batches.

public class CreateDatedConversionRateBatch implements Database.Batchable<sObject>,Database.Stateful,Database.AllowsCallouts {
    private string sessionId = null;
    public CreateDatedConversionRateBatch(string sessionIdParam) {
        // Need to use this quickly as it is likely to expire before the batch runs to completion, which is less than ideall
        sessionId = sessionIdParam;
    }
    
    public Database.QueryLocator start(Database.BatchableContext BC) {
        return Database.getQueryLocator('SELECT Id, IsoCode, ConversionRate FROM DatedConversionRate');
    }
    
    public void execute(Database.BatchableContext BC, List<DatedConversionRate> conversionList) {
        System.debug(LoggingLevel.INFO, 'executing batch');
        DateConversionRateCreate dcrc = new DateConversionRateCreate(sessionId);
        dcrc.create('GBP', 0.667400, DateTime.now());
    }
     
    public void finish(Database.BatchableContext BC) {
        System.debug(LoggingLevel.INFO, 'finish batch');
    }
}

Then start it with:

CreateDatedConversionRateBatch batchJob = new CreateDatedConversionRateBatch(UserInfo.getSessionId());
Id batchJobId = Database.executeBatch(batchJob, 1);
System.debug(batchJobId);

From the SerialBatchApexRangeChunkHandler debug log:

Do the same POST from Anonymous Apex with a Named Credential

I created a Named Credential, Connected App, and Auth Provider using the instructions in Using Named Credentials with the Apex Wrapper Salesforce Metadata API (apex-mdapi), they are pretty comprehensive on what is required.

Then I modified DateConversionRateCreate to handle the Session ID being null and to switch to the named credential in that scenario.

    Http h = new Http();
    HttpRequest req = new HttpRequest();
    req.setHeader('Content-Type', 'application/json');
    req.setBody(jsonstring);
    if(sessionId == null) {
        System.debug(LoggingLevel.INFO, 'Using Named Cred');
        req.setHeader('Authorization', 'OAuth {!$Credential.OAuthToken}');
        req.setEndpoint('callout:LocalRestAPI/services/data/v43.0/sobjects/DatedConversionRate');
    }
    else {
        req.setHeader('Authorization', 'OAuth ' + sessionId);
        req.setEndpoint('https://na87.salesforce.com/services/data/v43.0/sobjects/DatedConversionRate');
    }
    
    req.setMethod('POST');

The logs showed this worked and used the Named Credential for authentication.

Do the same POST from within a batch with a Named Credential

Test with:

CreateDatedConversionRateBatch batchJob = new CreateDatedConversionRateBatch(null);
Id batchJobId = Database.executeBatch(batchJob, 1);
System.debug(batchJobId);

Result:

So that is all working. The Named Credential (using OAuth and a refresh token) is providing a valid Session ID to use in a batch context. The callout is hitting the REST API and creating the new DateConversionRate.