要订阅RabbitMQ，首先需要安装RabbitMQ.Client包。可以使用NuGet进行安装。

然后，在ASP.NET Core Web API项目中，需要创建一个RabbitMQ的订阅者类。以下是一个简单的示例：

using RabbitMQ.Client;
using RabbitMQ.Client.Events;
using System;
using System.Text;

public class RabbitMQSubscriber
{
    private readonly IConnection _connection;
    private readonly IModel _channel;

    public RabbitMQSubscriber()
    {
        var factory = new ConnectionFactory() { HostName = "localhost" };
        _connection = factory.CreateConnection();
        _channel = _connection.CreateModel();
    }

    public void Subscribe(string queueName)
    {
        _channel.QueueDeclare(queue: queueName,
                              durable: false,
                              exclusive: false,
                              autoDelete: false,
                              arguments: null);

        var consumer = new EventingBasicConsumer(_channel);
        consumer.Received += (model, ea) =>
        {
            var body = ea.Body.ToArray();
            var message = Encoding.UTF8.GetString(body);
            Console.WriteLine("Received message: {0}", message);
        };

        _channel.BasicConsume(queue: queueName,
                              autoAck: true,
                              consumer: consumer);
    }

    public void CloseConnection()
    {
        _channel.Close();
        _connection.Close();
    }
}

在ASP.NET Core Web API的控制器中，可以调用RabbitMQSubscriber类来订阅消息。以下是一个示例：

using Microsoft.AspNetCore.Mvc;

[ApiController]
[Route("api/[controller]")]
public class RabbitMQController : ControllerBase
{
    private readonly RabbitMQSubscriber _subscriber;

    public RabbitMQController(RabbitMQSubscriber subscriber)
    {
        _subscriber = subscriber;
    }

    [HttpGet("subscribe")]
    public IActionResult Subscribe()
    {
        _subscriber.Subscribe("my_queue");
        return Ok("Subscribed to RabbitMQ");
    }

    // Other controller actions...
}

在Startup.cs文件中，需要将RabbitMQSubscriber类注册为服务。以下是一个示例：

using Microsoft.AspNetCore.Builder;
using Microsoft.AspNetCore.Hosting;
using Microsoft.Extensions.Configuration;
using Microsoft.Extensions.DependencyInjection;
using Microsoft.Extensions.Hosting;

public class Startup
{
    public Startup(IConfiguration configuration)
    {
        Configuration = configuration;
    }

    public IConfiguration Configuration { get; }

    public void ConfigureServices(IServiceCollection services)
    {
        services.AddSingleton();
        services.AddControllers();
    }

    public void Configure(IApplicationBuilder app, IWebHostEnvironment env)
    {
        if (env.IsDevelopment())
        {
            app.UseDeveloperExceptionPage();
        }

        app.UseRouting();

        app.UseEndpoints(endpoints =>
        {
            endpoints.MapControllers();
        });
    }
}

通过以上步骤，ASP.NET Core Web API就可以订阅RabbitMQ消息了。在控制器的订阅方法中，可以进行自定义的业务逻辑处理。