Example: Minor engine upgrade from 1.2.0.1 to 1.2.0.2

Find the DB cluster that you want to upgrade, and the template you used to create it. For example:

Description: Base Template to create Neptune Stack with Engine Version 1.2.0.1 using custom Parameter Groups
Parameters:
  DbInstanceType:
    Description: Neptune DB instance type
    Type: String
    Default: db.r5.large
Resources:
  NeptuneDBClusterParameterGroup:
    Type: 'AWS::Neptune::DBClusterParameterGroup'
    Properties:
      Family: neptune1.2
      Description: test-cfn-neptune-db-cluster-parameter-group-description
      Parameters:
        neptune_enable_audit_log: 0
  NeptuneDBParameterGroup:
    Type: 'AWS::Neptune::DBParameterGroup'
    Properties:
      Family: neptune1.2
      Description: test-cfn-neptune-db-parameter-group-description
      Parameters:
        neptune_query_timeout: 20000
  NeptuneDBCluster:
    Type: 'AWS::Neptune::DBCluster'
    Properties:
      EngineVersion: 1.2.0.1
      DBClusterParameterGroupName:
        Ref: NeptuneDBClusterParameterGroup
    DependsOn:
      - NeptuneDBClusterParameterGroup
  NeptuneDBInstance:
    Type: 'AWS::Neptune::DBInstance'
    Properties:
      DBClusterIdentifier:
        Ref: NeptuneDBCluster
      DBInstanceClass:
        Ref: DbInstanceType
      DBParameterGroupName:
        Ref: NeptuneDBParameterGroup
    DependsOn:
      - NeptuneDBCluster
      - NeptuneDBParameterGroup
Outputs:
  DBClusterId:
    Description: Neptune Cluster Identifier
    Value:
      Ref: NeptuneDBCluster

Update the EngineVersion property from 1.2.0.1 to 1.2.0.2:

Description: Template to upgrade minor engine version to 1.2.0.2
Parameters:
  DbInstanceType:
    Description: Neptune DB instance type
    Type: String
    Default: db.r5.large
Resources:
  NeptuneDBClusterParameterGroup:
    Type: 'AWS::Neptune::DBClusterParameterGroup'
    Properties:
      Family: neptune1.2
      Description: test-cfn-neptune-db-cluster-parameter-group-description
      Parameters:
        neptune_enable_audit_log: 0
  NeptuneDBParameterGroup:
    Type: 'AWS::Neptune::DBParameterGroup'
    Properties:
      Family: neptune1.2
      Description: test-cfn-neptune-db-parameter-group-description
      Parameters:
        neptune_query_timeout: 20000
  NeptuneDBCluster:
    Type: 'AWS::Neptune::DBCluster'
    Properties:
      EngineVersion: 1.2.0.2
      DBClusterParameterGroupName:
        Ref: NeptuneDBClusterParameterGroup
    DependsOn:
      - NeptuneDBClusterParameterGroup
  NeptuneDBInstance:
    Type: 'AWS::Neptune::DBInstance'
    Properties:
      DBClusterIdentifier:
        Ref: NeptuneDBCluster
      DBInstanceClass:
        Ref: DbInstanceType
      DBParameterGroupName:
        Ref: NeptuneDBParameterGroup
    DependsOn:
      - NeptuneDBCluster
      - NeptuneDBParameterGroup
Outputs:
  DBClusterId:
    Description: Neptune Cluster Identifier
    Value:
      Ref: NeptuneDBCluster

Now use AWS CloudFormation to run the revised template.