Which strategies exist to monetize a small following of 1000 people on Instagram?

Instagram is great for attracting targeted customers. But the biggest problem is you can't add links in the post. The links are added as text and are not clickable. I think following is the only best way to covert your followers to customers:

1. Keep posting quality content. Always post unique, catchy and "can't resist to share" content. Content must force its viewer to share. And by force I mean it must be AMAZINNNGGG.... It's not like you just wrote "share this or you will go to hell" :D

2. Use hashtags. Research about them. Use most used hastags like "followme" "instacool" etc. But use that are more relevant to what you are selling. When you type "#" inside description of your post, Instagram will suggest you top hashtags. Select the one that is relevant to your business.

3. Give your followers free content like Preview PDF book (or something else that introduce your offer) in the website link on your profile page.

4. With each post add a line like "Click the link in the bio to get the free book" or "To know more click the link in the bio" or something else.

5. When user clicks the link you must take him to a page where he/she enters his/her email and then taken to a FREE content or download.

6. You collected their emails. Now its time to send newsletters about your paid content. Send them in 3 days gap or weekly. Don't send too much mails otherwise they will report your email as spam.

This is how I made a good business from Instagram. I hope it helps you too. :)

You want to grab their attention quickly.
Instagram obviously is a photo or quick video platform so that photo or video needs to grab a users attention immediately and make them want to either click on your business link or reply asking for more information etc.
Instagram can be a tough way to monetize in comparison to other Social Media platforms, but if done successfully it's definitely doable!!

To monetise anything you need
1 - an audience
2 - a product to sell to the audience

Your question is not clear on either of these points. So first you need to answer these questions:

Are the people in a single industry / areas of interest?
Where do other people who like that industry / interest area hang out?
What brands do they like?
Can you offer the brand access to your audience for a fee?
What add-on or associated services can you combine with your core product?
Consider the customer life-time - from newbie to leaving the brand - what are the stages you go through (CRM)?
What products and services can you offer that align with each stage and how can you help the brand stay relevant to the audience through each of these stages?

Go to work and write up your answers..... then we can give you more tasks.