Yes you can, and there are several options to consider. G-suite is the normal option for mission critical email. The price can change but it's normally about $5 a month per email. Our most popular and recommended service is WorkSpace at EZ Domain Name which starts at $2.99/mo for one email, or $5.99/mo for 5 emails with IMAP. We use EZ Domain Name ourselves for mission critical email.
It's possible if you are having email issues that you just need to add SPF, DKIM, and DMARC to your DNS. SPF and DKIM is already added to all shared and reseller hosting websites.
These services are also popular for sending email remotely.
Sendgrid, Postmark, Mailgun, Mailjet, Sparkpost, Sendinblue, Mandrill(Mailchimp) and AmazonSES.
Prices and options do vary, so contacting them directly is the best way to compare which is best for you. To use a remote email service you will just change the MX record in your DNS section of the website's cPanel.
To use these services you just redirect your MX record to them in your cPanel or add a ticket and we will do this for you.