Appleton signed 10 new players in the days leading up to their League One opener, but Pompey looked a cohesive unit.
They took the lead after 18 minutes when Cherries goalkeeper Shwan Jalal let Izale McLeod's tame, long-range effort squirm through his legs, but Bournemouth bounced back after the break and Lee Barnard headed them level from Simon Francis's cross after 78 minutes.
Appleton said: "I'm going to be pushing very hard. We've got [Gibraltar international] Liam Walker and hopefully his international clearance will be ready for Tuesday.
"Jordan Obita will be joining us on loan from Reading on Monday, but that only gets us to 17. We need at least another three players to be competitive. If we pick an injury up, we're snookered."
Appleton added: "The phrase '24/7' doesn't even come into what we've had to do over the last few days. I'm not sure how much sleep I've had and that's not through worrying; that's just not being able to have enough time in the day to get things done."