The first change that I would do is to remove the async. It will just mess out the code with unneeded Promises.
Second, I removed the need to catch the exception, verifying the existence of the file with fs.existsSync(path). Try to not to rise exceptions as often as possible. If you know something can rise an exception, test it.
Last, and most important, I created a reading stream of the file and piped the result to the response with fs.createReadStream(path).pipe(res). This way, the client recieves the file as it is read and your memory is spared. Great for large files.
Reading a file can be memory intensive, so loading it all in memory is a bad practice. You just need a handfull of request to overload your machine.
You can read more on the pipe method here.
In this example, any GET call to /router/show_file will return the pdf.
const express = require('express')
const app = express()
const fs = require('fs')
const router = express.Router()
router.get('/show_file', (req, res) => {
    const path = './public/images/1.pdf'
    if (fs.existsSync(path)) {
        res.contentType("application/pdf");
        fs.createReadStream(path).pipe(res)
    } else {
        res.status(500)
        console.log('File not found')
        res.send('File not found')
    }
})
app.use('/router', router) // Here we pass the router to the app with a path
app.listen(9999, () => console.log('Listening to port 9999'))